Differentiation
Class 12th Mathematics RS Aggarwal Solution
- sin 4x Differentiate each of the following w.r.t. x:
- cos 5x Differentiate each of the following w.r.t. x:
- tan 3x Differentiate each of the following w.r.t. x:
- cos x3 Differentiate each of the following w.r.t. x:
- cot2x Differentiate each of the following w.r.t. x:
- tan3x Differentiate each of the following w.r.t. x:
- cotroot {x} Differentiate each of the following w.r.t. x:
- root {tanx} Differentiate each of the following w.r.t. x:
- (5 + 7x)6 Differentiate each of the following w.r.t. x:
- (3 – 4x )5 Differentiate each of the following w.r.t. x:
- (2x2 – 3x +4)5 Differentiate each of the following w.r.t. x:
- (𝔞𝓍2 + 𝔟𝓍 + c )6 Differentiate each of the following w.r.t. x:…
- {1}/{ ( x^{2} - 3x+5 ) ^{3} } Differentiate each of the following w.r.t. x:…
- root { { a^{2} - x^{2} }/{ a^{2} + x^{2} } } Differentiate each of the…
- root { {1+sinx}/{1-sinx} } Differentiate each of the following w.r.t. x:…
- cos2𝓍3 Differentiate each of the following w.r.t. x:
- sec3 (𝓍2+1) Differentiate each of the following w.r.t. x:
- root {cos3x} Differentiate each of the following w.r.t. x:
- cube root {sin2x} Differentiate each of the following w.r.t. x:
- root {1+cotx} Differentiate each of the following w.r.t. x:
- cosec^{3} {1}/{ x^{2} } Differentiate each of the following w.r.t. x:…
- root { sinx^{3} } Differentiate each of the following w.r.t. x:
- root {xsinx} Differentiate each of the following w.r.t. x:
- root { cotsqrt{x} } Differentiate each of the following w.r.t. x:…
- cot3𝓍2 Differentiate each of the following w.r.t. x:
- cos ( sinroot {9x+b} ) Differentiate each of the following w.r.t. x:…
- root { cosec ( x^{3} + 1 ) } Differentiate each of the following w.r.t. x:…
- sin 5𝓍 cos 3𝓍 Differentiate each of the following w.r.t. x:
- sin 2𝓍 sin 𝓍 Differentiate each of the following w.r.t. x:
- cos 4𝓍 cos 2𝓍 Differentiate each of the following w.r.t. x:
- Find {dy}/{dx} , when:𝒴 =sin ( { 1+x^{2} }/{ 1-x^{2} } ) Find ,…
- Find {dy}/{dx} , when:𝒴 = { ( sinx+x^{2} ) }/{cot2x} Find , when:…
- If 𝒴 = { (cosx-sinx) }/{ (cosx+sinx) } , prove that {dy}/{dx} + 𝒴2…
- If 𝒴 = { (cosx+sinx) }/{ (cosx-sinx) } , prove that {dy}/{dx} =sec2…
- Find the second derivate of :(i) x11 (ii) 5x(iii) tan x (iv) cos-1x…
- Find the second derivative of:(i) x sin x(ii) e2x cos 3x(iii) x3 log x…
- If 𝒴 = x + tan x, show that cos2 . {d^{2}y}/{ dx^{2} } - 2𝒴+ 2x = 0.…
- If 𝒴 = 2 sin x + 3 cos x, s how that 𝒴 + {d^{2}y}/{ dx^{2} } =0.…
- If 𝒴 = 3 cos (log x) + 4 sin (log x), prove that x2𝒴2 + x𝒴1 + 𝒴 = 0.…
- If y = e- cos x , show that {d^{2}y}/{ dx^{2} } = 2e-xsin x .…
- If 𝒴 = sec x – tan x. show that (cos x) {d^{2}y}/{ dx^{2} } = 𝒴2.…
- If 𝒴 = (cosec x + cot x), prove that (sin x) {d^{2}y}/{ dx^{2} } - 𝒴2 =…
- If 𝒴 = tan-1 , show that (1 + x2) {d^{2}y}/{ dx^{2} } + 2x {dy}/{dx}…
- If 𝒴 = sin (sin x), prove that {d^{2}y}/{ dx^{2} } + (tan )…
- If 𝒴 = a cos (log x) + b sin (log x), prove that x2𝒴2 + x𝒴1 + 𝒴 = 0.…
- Find the second derivative of e3x sin 4x .
- Find the second derivative of sin 3 x cos 5x.
- If 𝒴 = etan , prove that (cos2x) {d^{2}y}/{ dx^{2} } - (1+ sin 2x).…
- If 𝒴 = {logx}/{x} , show that {d^{2}y}/{ dx^{2} } = { (2logx-3)…
- If 𝒴 = eax cos bx, show that {d^{2}y}/{ dx^{2} } - 2a {dy}/{dx} +…
- If 𝒴 = ea cos-1x, - 1 ≤ x ≤ 1, show that (1 - x2) {d^{2}y}/{ dx^{2} } -…
- If x = at2 and 𝒴 = 2at, find {d^{2}y}/{ dx^{2} } at t = 2.…
- If x = a(θ – sin θ ) and 𝒴 = a(1 – cos θ), find {d^{2}y}/{ dx^{2} } at θ…
- If 𝒴 = sin (log x), prove that x2 {d^{2}y}/{ dx^{2} } +x {dy}/{dx}…
- If 𝒴 = {sin^{-1}x}/{ root { 1-x^{2} } } , show that (1 - x2)…
- If 𝒴 = ex sin x, prove that {d^{2}y}/{ dx^{2} } - 2 {dy}/{dx} + 2𝒴…
- If x = a ( costheta +logtan { theta }/{2} ) and 𝒴 = a sin θ, show that…
- If x = cos t + log tan {t}/{2} , 𝒴 = sin t then find the values of…
- If 𝒴 = xx, prove that {d^{2}y}/{ dx^{2} } - {1}/{y} ( {dy}/{dx}…
- If 𝒴 = (cot-1 )2, then show that (x2 + 1)2 {d^{2}y}/{ dx^{2} } + 2x (x2 +…
- If 𝒴 = { x + root { x^{2} + 1 } } ^{m} , then show that (x2 + 1)…
- If 𝒴 = log [x + [x + root { x^{2} + a^{2} } ] , then prove that…
- If x = a(cos θ + θ sin θ ) and 𝒴 = a(sin θ - θ cos θ), show that…
- If x = a cos θ + b sin θ and 𝒴 = a sin θ – b cos θ , show that y^{2}…
- (i) e^{4x} (ii) e^{-5x} (iii) (e)^ { x^{3} } Differentiate each of the…
- (i) e^{2/x} (ii) e^ { root {x} } (iii) e^ { - 2 root {x} } Differentiate…
- (i) e^{cotx} (ii) e^{-sin2x} (iii) e^ { root {sinx} } Differentiate each of…
- (i) tan (log x)(ii) log (sec x)(iii) log (sin (x/2)) Differentiate each of the…
- (i)log3 x(ii) 2^{-x} (iii) 3^{x+2} Differentiate each of the following…
- (i) log ( x + {1}/{x} ) (ii) log (sin (3x))(iii) log ( x + root { 1+x^{2}…
- e^ { root {x} } logx Differentiate each of the following w.r.t. x:…
- logsinroot { x^{2} + 1 } Differentiate each of the following w.r.t. x:…
- e2x sin 3x Differentiate each of the following w.r.t. x:
- e3x cos 2x Differentiate each of the following w.r.t. x:
- e-5x cot 4x Differentiate each of the following w.r.t. x:
- ex log (sin 2x) Differentiate each of the following w.r.t. x:
- log (cosecx-cotx) Differentiate each of the following w.r.t. x:
- log ( sec {x}/{2} + tan frac {x}/{2} ) Differentiate each of the following…
- root { { 1+e^{x} }/{ 1-e^{x} } } Differentiate each of the following w.r.t.…
- { e^{x} + e^{-x} }/{ e^{x} - e^{-x} } Differentiate each of the following…
- xe^ { root {sinx} } Differentiate each of the following w.r.t. x:…
- e^{sinx} sin ( e^{x} ) Differentiate each of the following w.r.t. x:…
- e^ { root { 1-x^{2} } } tanx Differentiate each of the following w.r.t. x:…
- { e^{x} }/{1+cosx} Differentiate each of the following w.r.t. x:
- x^{3}e^{x}cosx Differentiate each of the following w.r.t. x:
- e^{x}cosx Differentiate each of the following w.r.t. x:
- cos^{-1}2x Differentiate each of the following w.r.t. x:
- tan^{-1}x^{2} Differentiate each of the following w.r.t. x:
- sec^{-1}root {x} Differentiate each of the following w.r.t. x:
- sin^{-1} {x}/{a} Differentiate each of the following w.r.t. x:
- tan^{-1} (logx) Differentiate each of the following w.r.t. x:
- cot^{-1} ( e^{x} ) Differentiate each of the following w.r.t. x:
- log (tan^{-1}x) Differentiate each of the following w.r.t. x:
- cot^{-1}x^{3} Differentiate each of the following w.r.t. x:
- sin^{-1} (cosx) Differentiate each of the following w.r.t. x:
- ( 1+x^{2} ) tan^{-1}x Differentiate each of the following w.r.t. x:…
- tan^{-1} (cotx) Differentiate each of the following w.r.t. x:
- log ( sin^{-1}x^{4} ) Differentiate each of the following w.r.t. x:…
- ( cot^{-1}x^{2} ) ^{3} Differentiate each of the following w.r.t. x:…
- tan^{-1} ( cosroot {x} ) Differentiate each of the following w.r.t. x:…
- tan (sin^{-1}x) Differentiate each of the following w.r.t. x:
- e^ { tan^{-1}root {x} } Differentiate each of the following w.r.t. x:…
- root { sin^{-1}x^{2} } Differentiate each of the following w.r.t. x:…
- If y = sin^{-1} (cosx) + cos^{-1} (sinx) prove that {dy}/{dx} = - 2 .…
- Prove that {d}/{dx} { 2xtan^{-1}x-log ( 1+x^{2} ) } = 2tan^{-1}x .…
- sin^{-1} { root { {1-cosx}/{2} } } Differentiate each of the following w.r.t…
- tan^{-1} ( {sinx}/{1+cosx} ) Differentiate each of the following w.r.t x:…
- cot^{-1} ( {1+cosx}/{sinx} ) Differentiate each of the following w.r.t x:…
- cot^{-1} ( root { {1+cosx}/{1-cosx} } ) Differentiate each of the following…
- tan^{-1} ( {cosx+sinx}/{cosx-sinx} ) Differentiate each of the following…
- cot^{-1} ( {cosx-sinx}/{cosx+sinx} ) Differentiate each of the following…
- cot^{-1} ( root { {1+cos3x}/{1-cos3x} } ) Differentiate each of the…
- sec^{-1} ( {1+tan^{2}x}/{1-tan^{2}x} ) Differentiate each of the following…
- sin^{-1} ( {1-tan^{2}x}/{1+tan^{2}x} ) Differentiate each of the following…
- cosec^{-1} ( {1+tan^{2}x}/{2tanx} ) Differentiate each of the following…
- cot^{-1} (cosecx+cotx) Differentiate each of the following w.r.t x:…
- tan^{-1} (cotx) + cot^{-1} (tanx) Differentiate each of the following w.r.t x:…
- sin^{-1} { root { 1-x^{2} } } Differentiate each of the following w.r.t x:…
- sin^{-1} ( root { {1-x}/{2} } ) Differentiate each of the following w.r.t…
- cos^{-1} { root { {1+x}/{2} } } Differentiate each of the following w.r.t…
- cos^{-1} { root { 1-x^{2} } } Differentiate each of the following w.r.t x:…
- sin^{-1} { 2x root { 1-x^{2} } } Differentiate each of the following w.r.t x:…
- sin^{-1} ( 3x-4x^{3} ) Differentiate each of the following w.r.t x:…
- sin^{-1} ( 1-2x^{2} ) Differentiate each of the following w.r.t x:…
- sec^{-1} ( {1}/{ root { 1-x^{2} } } ) Differentiate each of the following…
- tan^{-1} ( {x}/{ root { 1-x^{2} } } ) Differentiate each of the following…
- tan^{-1} ( {x}/{ 1 + root { 1-x^{2} } } ) Differentiate each of the…
- cot^{-1} ( { root { 1-x^{2} } }/{x} ) Differentiate each of the following…
- sec^{-1} ( {1}/{ 1-2x^{2} } ) Differentiate each of the following w.r.t x:…
- sin^{-1} { {1}/{ root { 1+x^{2} } } } Differentiate each of the following…
- tan^{-1} ( {1+x}/{1-x} ) Differentiate each of the following w.r.t x:…
- cot^{-1} ( {1+x}/{1-x} ) Differentiate each of the following w.r.t x:…
- tan^{-1} ( { 3x-x^{3} }/{ 1-3x^{2} } ) Differentiate each of the following…
- cosec^{-1} ( { 1+x^{2} }/{2x} ) Differentiate each of the following w.r.t x:…
- sec^{-1} ( { 1+x^{2} }/{ 1-x^{2} } ) Differentiate each of the following…
- sin^{-1} ( {1}/{ root { 1+x^{2} } } ) Differentiate each of the following…
- sec^{-1} ( { x^{2} + 1 }/{ x^{2} - 1 } ) Differentiate each of the following…
- cos^{-1} ( { 1-x^{2n} }/{ 1+x^{2n} } ) Differentiate each of the following…
- tan^{-1} { {x}/{ root { a^{2} - x^{2} } } } Differentiate each of the…
- sin^{-1} { 2ax root { 1-a^{2}x^{2} } } Differentiate each of the following…
- tan^{-1} { { root { 1+a^{2}x^{2} } - 1 }/{ax} } Differentiate each of the…
- sin^{-1} { { x^{2} }/{ root { x^{4} + a^{4} } } } Differentiate each of the…
- tan^{-1} { { e^{2x} + 1 }/{ e^{2x} - 1 } } Differentiate each of the…
- cos^{-1} (2x) + 2cos^{-1}root { 1-4x^{2} } Differentiate each of the…
- tan^{-1} ( {a-x}/{1+ax} ) Differentiate each of the following w.r.t x:…
- tan^{-1} { { root {x}-x }/{1+x^{3/2}} } Differentiate each of the following…
- tan^{-1} ( { root {a} + sqrt{x} }/{ 1 - sqrt{ax} } ) Differentiate each of…
- tan^{-1} ( {3-2x}/{1+6x} ) Differentiate each of the following w.r.t x:…
- tan^{-1} ( {5x}/{ 1-6x^{2} } ) Differentiate each of the following w.r.t x:…
- tan^{-1} ( {2x}/{ 1+15x^{2} } ) Differentiate each of the following w.r.t x:…
- If t = tan^{-1} ( {ax-b}/{bx+a} ) , prove that {dy}/{dx} = frac {1}/{…
- If y = sin^{-1} ( {2x}/{ 1+x^{2} } ) + sec^{-1} ( frac { 1+x^{2} }/{…
- If y = sec^{-1} ( {x+1}/{x-1} ) + sin^{-1} ( frac {x-1}/{x+1} ) , show…
- If y = sin { 2tan^{-1} ( root { {1-x}/{1+x} } ) } , show that…
- If y = tan^{-1} { { root {1+x} - sqrt{1-x} }/{ sqrt{1+x} + sqrt{1-x} } } .…
- Differentiate sin^{-1} ( { 2^{x+1} }/{ 1+4^{x} } ) w. r. t. x…
- x2 + y2 = 4 Find , when:
- { x^{2} }/{ a^{2} } + frac { y^{2} }/{ b^{2} } = 1 Find , when:
- root {x} + sqrt{y} = sqrt{a} Find , when:
- x^{2/3}+y^{2/3} = a^{2/3} Find , when:
- xy = c2 Find , when:
- x2 + y2 — 3xy = 1 Find , when:
- xy2 — x2y — 5 = 0 Find , when:
- (x2 + y2)2 = xy Find , when:
- x2 + y2 = log (xy) Find , when:
- xn + yn = an Find , when:
- x sin 2y = y cos 2x Find , when:
- sin2x + 2cos y + xy Find , when:
- y sec x + tan x + x2y = 0 Find , when:
- cot (xy) + xy = y Find , when:
- y tan x — y2 cos x + 2x = 0 Find , when:
- ex log y = sin—1 x + sin —1y Find , when:
- xy log (x + y) = 1 Find , when:
- tan (x + y) + tan (x — y) = 1 Find , when:
- logroot { x^{2} + y^{2} } = tan^{-1} {y}/{x} Find , when:
- If y = x sin y , prove that ( x c. {dy}/{dx} ) = frac {y}/{ (1-xcosy) }…
- If xy = tan (xy), show that {dy}/{dx} = frac {-y}/{x} . Find , when:…
- If y log x = (x — y), prove that {dy}/{dx} = frac {logx}/{ (1+logx)^{2} }…
- If cos y = x cos (y + a), prove that {dy}/{dx} = frac { cos^{2} (y+a)…
- If cos^{-1} ( { x^{2} - y^{2} }/{ x^{2} + y^{2} } ) = tan^{-1}a , prove…
- Find {dy}/{dx} , when: y = x^{1/x} Find , when:
- Find {dy}/{dx} , when: y = x^ { root {x} } Find , when:
- Find {dy}/{dx} , when: y = (logx)^{x} Find , when:
- Find {dy}/{dx} , when: y = x^{sinx} Find , when:
- Find {dy}/{dx} , when: y = x^ { (cos^{-1}x) } Find , when:
- Find {dy}/{dx} , when: y = (tanx)^{1/x} Find , when:
- Find {dy}/{dx} , when: y = (sinx)^{cosx} Find , when:
- Find {dy}/{dx} , when: y = (logx)^{sinx} Find , when:
- Find {dy}/{dx} , when: y = (cosx)^{logx} Find , when:
- Find {dy}/{dx} , when: y = (tanx)^{sinx} Find , when:
- Find {dy}/{dx} , when: y = (cosx)^{cosx} Find , when:
- Find {dy}/{dx} , when: y = (tanx)^{cotx} Find , when:
- Find {dy}/{dx} , when: y = x^{sin2x} Find , when:
- Find {dy}/{dx} , when: y = (sin^{-1}x)^{x} Find , when:
- Find {dy}/{dx} , when: y = sin ( x^{x} ) Find , when:
- Find {dy}/{dx} , when: y = (3x+5)^ { (2x-3) } Find , when:
- Find {dy}/{dx} , when: y = (x+1)^{3} (x+2)^{4} (x+3)^{5} Find , when:…
- Find {dy}/{dx} , when: y = root { { (x-1) (x-2) }/{ (x-3) (x-4) (x-5)…
- Find {dy}/{dx} , when: y = (2-x)^{3} (3+2x)^{5} Find , when:…
- Find {dy}/{dx} , when: y = cosxcos2xcos3x Find , when:
- Find {dy}/{dx} , when: y = { x^{5}root {x+4} }/{ (2x+3)^{2} } Find ,…
- Find {dy}/{dx} , when: y = { (x+1)^{2}root {x-1} }/{ (x+4)^{3} c.…
- Find {dy}/{dx} , when: y = { root {x} (3x+5)^{2} }/{ sqrt{x+1} } Find ,…
- Find {dy}/{dx} , when: y = { x^{2}root {1+x} }/{ ( 1+x^{2} ) ^{3/2} }…
- Find {dy}/{dx} , when: y = root { (x-2) (2x-3) (3x-4) } Find , when:…
- Find {dy}/{dx} , when: y = sin2xsin3xsin4x Find , when:
- Find {dy}/{dx} , when: y = {x^{3}sinx}/{ e^{x} } Find , when:…
- Find {dy}/{dx} , when: y = {e^{x}logx}/{ x^{2} } Find , when:…
- Find {dy}/{dx} , when: y = {xcos^{-1}x}/{ root { 1-x^{2} } } Find ,…
- Find {dy}/{dx} , when: y = (1+x) ( 1+x^{2} ) ( 1+x^{4} ) ( 1+x^{6} ) Find…
- Find {dy}/{dx} , when: y = x^{x} - 2^{sinx} Find , when:
- Find {dy}/{dx} , when: y = (logx)^{x} + x^{logx} Find , when:…
- Find {dy}/{dx} , when: y = x^{sinx} + (sinx)^{cosx} Find , when:…
- Find {dy}/{dx} , when: y = (xcosx)^{x} + (xsinx)^{1/x} Find , when:…
- Find {dy}/{dx} , when: y = (sinx)^{x} + sin^{-1}root {x} Find , when:…
- Find {dy}/{dx} , when: y = x^{xcosx} + ( { x^{2} + 1 }/{ x^{2} - 1 } )…
- Find {dy}/{dx} , when: y = e^{x}sin^{3}xcos^{4}x Find , when:…
- Find {dy}/{dx} , when: y = 2^{x} c. e^{3x}sin4x Find , when:…
- Find {dy}/{dx} , when: y = x^{x} c. e^ { (2x+5) } Find , when:…
- Find {dy}/{dx} , when: y = (2x+3)^{5} (3x-5)^{7} (5x-1)^{3} Find , when:…
- Find {dy}/{dx} , when: (cosx)^{y} = (cosy)^{x} Find , when:…
- Find {dy}/{dx} , when: (tanx)^{y} = (tany)^{x} Find , when:…
- Find {dy}/{dx} , when: y = ( (logx)^{x} + (x)^{logx} Find , when:…
- If y = {sin^{-1}x}/{ root { 1-x^{2} } } , prove that ( 1-x^{2} )…
- If y = root {x+y} , prove that {dy}/{dx} = frac {1}/{ (2y-1) } .…
- If x^{a}y^{b} = (x+y)^ { (a+b) } , prove that {dy}/{dx} = frac {y}/{x}…
- If ( x^{x} + y^{x} ) = 1 , show that {dy}/{dx} = - { frac { x^{x}…
- If y = e^{sinx} + (tanx)^{x} , prove that {dy}/{dx} = e^{sinx} cosx +…
- If y = log ( x + root { 1+x^{2} } ) , prove that {dy}/{dx} = frac {1}/{…
- If y = logsinroot { x^{2} + 1 } , prove that {dy}/{dx} = frac {…
- If y = logroot { {1-cosx}/{1+cosx} } , show that {dy}/{dx} = cosecx…
- If y = logtan ( { pi }/{4} + frac {x}/{2} ) , show that {dy}/{dx} =…
- If y = root { {1-sin2x}/{1+sin2x} } , show that {dy}/{dx} + sec^{2} (…
- If y = logroot { {1+cos^{2}x}/{ 1-e^{2x} } } , show that {dy}/{dx} =…
- If y = (x)^{cosx} + (sinx)^{tanx} , prove that {dy}/{dx} = x^{cosx} {…
- If y = (sinx)^{cosx} + (cosx)^{sinx} , prove tha {dy}/{dx} = (sinx)^{cosx}…
- If y = (tanx)^{cotx} + (cotx)^{tanx} , {dy}/{dx} = (tanx)^{cotx} c.…
- If y = x^{cosx} + (cosx)^{x} , prove that {dy}/{dx} = x^{cosx} c. {…
- If y = x^{logx} + (logx)^{x} prove that {dy}/{dx} = x^ { (logx) } { frac…
- If y-x^ { ( x^{2} - 3 ) } + (x-3)^ { x^{2} } , find {dy}/{dx} .…
- If f (x) = ( {3+x}/{1+x} ) ^{2+3x} , find f^ { there eξ sts } (0) .…
- If y = (sinx)^{x} + sin^{-1}root {x} find {dy}/{dx} .
- If ( x^{2} + y^{2} ) ^{2} = xy , find {dy}/{dx} .
- y = x^{cotx} + { 2x^{2} - 3 }/{ x^{2} + x+2 } , find {dy}/{dx} .…
- If y = tan^{-1} {a}/{x} + logroot { frac {x-a}/{x+a} } , prove that…
- If , prove that {dy}/{dx} = frac {y}/{x} .
- If , prove that {dy}/{dx} = frac {y^{2}cotx}/{ (1-ylogsinx) }
- If y = (cosx)^ { (cosx) } (cosx) l. s infinity , prove that {dy}/{dx} = frac…
- If y = root { x + sqrt { x + sqrt { x + l. s infinity } } } , prove that {dy}/{dx}…
- If y = root { cosx + sqrt { cosx + sqrt{cosx} + l. s infinity } } , prove that…
- If y = root { tanx + sqrt { tanx + sqrt { tanx + l. s infinity } } } prove that…
- If y = root { logx + sqrt { logx + sqrt { logx + l. s infinity } } } , show that…
- If y = a^ { a^{x}l. s infinity } , prove that {dy}/{dx} = frac { y^{2} (logy) }/{…
- If prove that {dy}/{dx} = frac {y}/{ (2y-x) }
- Differentiate x^{6} with respect to ( 1 / root {x} )
- Differentiate log with respect to
- Differentiate e^{sinx} with respect to cosx .
- Differentiate tan^{-1}root { { 1-x^{2} }/{ 1+x^{2} } } with respect to…
- Differentiate tan^{-1} ( {2x}/{ 1-x^{2} } ) with respect to sin^{-1} ( {2x}/{…
- Differentiate tan^{-1} ( {x}/{ root { 1-x^{2} } } ) with respect to cos^{-1} (…
- Differentiate sin^{3}x with respect to cos^{3}x
- Differentiate cos^{-1} ( { 1-x^{2} }/{ 1+x^{2} } ) with respect to tan^{-1} ( {…
- Differentiate tan^{-1} ( { root { 1+x^{2} } - 1 }/{x} ) with respect to sin^{-1} (…
- Differentiate tan^{-1} ( { root { 1-x^{2} } }/{x} ) with respect to cos^{-1} ( 2x…
- Find {dy}/{dx} , when𝒳 = at2, 𝒴 = 2at Find , when
- Find {dy}/{dx} , when𝒳 = a cos θ , 𝒴 = b sin θ Find , when…
- Find {dy}/{dx} , when𝒳 = a cos2θ , 𝒴 = b sin2θ Find , when…
- Find {dy}/{dx} , when𝒳 = a cos3θ, 𝒴 = a sin3θ Find , when
- Find {dy}/{dx} , when𝒳 = a(1 – cos θ), 𝒴 = a(θ + sin θ) Find , when…
- Find {dy}/{dx} , when𝒳 = a log t, 𝒴 = b sin t Find , when
- Find {dy}/{dx} , when𝒳 = (log t + cos t), 𝒴 = (et + sin t) Find , when…
- Find {dy}/{dx} , when𝒳 = cos θ + cos 2θ, 𝒴 = sin θ + sin 2θ Find , when…
- Find {dy}/{dx} , when𝒳 = root {sin2theta } , 𝒴 = root {cos2theta }…
- Find {dy}/{dx} , when𝒳 = eθ (sin θ + cos θ), 𝒴 = eθ (sin θ - θ cos θ)…
- Find {dy}/{dx} , when𝒳 = a (cos θ + θ sin θ ), 𝒴 = a (sin θ - θ cos θ)…
- Find {dy}/{dx} , when𝒳 = {3at}/{ ( 1+t^{2} ) } , 𝒴 = { 3at^{2}…
- Find {dy}/{dx} , when𝒳 = { 1-t^{2} }/{ 1+t^{2} } , 𝒴 = {2t}/{…
- Find {dy}/{dx} , when𝒳 = cos-1 {1}/{ root { 1+t^{2} } } , 𝒴 =…
- If 𝒳 = 2 cos t – 2 cos3t, 𝒴 = sin t – 2 sin3t, show that {dy}/{dx} =cot…
- If 𝒳 = {1+logt}/{ t^{2} } and 𝒴 = {3+2logt}/{t} {dy}/{dx} =t.…
- If 𝒳 = a(θ – sin θ), 𝒴 = a(1 – cos θ), find {dy}/{dx} at θ = { pi…
- If 𝒳 = 2 cosθ – cos 2θ and 𝒴 = 2 sin θ – sin 2θ, show that {dy}/{dx}…
- If 𝒳 = {sin^{3}t}/{ root {cos2t} } , 𝒴 = {cos^{3}t}/{ root {cos2t} } , find…
- If 𝒳 = (2 cos θ – cos 2θ) and = (2sin θ – sin 2θ), find ( {d^{2}y}/{ dx^{2} } ) _ {…
- If 𝒳 = a (θ – sin θ), 𝒴 = a( 1+cos θ), find {d^{2}y}/{ dx^{2} } .…
Exercise 10a
Question 1.Differentiate each of the following w.r.t. x:
sin 4x
Answer:
Formulae:
• 
• 
Let,
y = sin 4x
and u = 4x
therefore, y = sin u
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
= cos 4x . 4
= 4 cos 4x
Question 2.
Differentiate each of the following w.r.t. x:
cos 5x
Answer:
Formulae:
• 
•
Let,
y = cos 5x
and u = 5x
therefore, y= cos u
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= - sin 5x . 5
= - 5 sin 5x
Question 3.
Differentiate each of the following w.r.t. x:
tan 3x
Answer:
Formulae:
• 
•
Let,
y = tan 3x
and u = 3x
therefore, y= tan u
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= sec2 3x . 3
= 3 sec2 3x
Question 4.
Differentiate each of the following w.r.t. x:
cos x3
Answer:
Formulae:
•
• 
Let,
y = cos x3
and u = x3
therefore, y= cos u
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= - sin x3 . 3x2
= - 3x2 sin x3
Question 5.
Differentiate each of the following w.r.t. x:
cot2x
Answer:
Formulae:
• 
•
Let,
y = cot2 x
and u = cot x
therefore, y= u2
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= 2 cot x .(- cosec2 x)
= - 2cot x . cosec2 x
Question 6.
Differentiate each of the following w.r.t. x:
tan3x
Answer:
Formulae:
•
•
Let,
y = tan3 x
and u = tan x
therefore, y= u3
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= 3 tan2 x .(sec2 x)
= 3 tan2 x . sec2 x
Question 7.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
• 
Let,
and 
therefore, y= cot u
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 8.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
• 
Let,
and u = tan x
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 9.
Differentiate each of the following w.r.t. x:
(5 + 7x)6
Answer:
Formulae:
•
•
• 
• 
Let,
y = (5+7x)6
and u = (5+7x)
therefore, y = u6
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= 6. (5+7x)5. (0+7) …………. 
= 42. (5+7x)5
Question 10.
Differentiate each of the following w.r.t. x:
(3 – 4x )5
Answer:
Formulae:
•
•
• 
•
Let,
y = (3-4x)5
and u = (3-4x)
therefore, y = u5
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
= 5. (3-4x)4. (0-4) ………….
= -20 (3-4x)4
Question 11.
Differentiate each of the following w.r.t. x:
(2x2 – 3x +4)5
Answer:
Formulae:
•
•
•
•
Let,
y = (2x2 – 3x + 4)5
and u = (2x2 – 3x + 4)
therefore, y = u5
Differentiating above equation w.r.t. x,
………… By chain rule
………….
= 5. (2x2 – 3x + 4)4. (4x-3+0) …………. 
= 5. (2x2 – 3x + 4)4 (4x-3)
Question 12.
Differentiate each of the following w.r.t. x:
(𝔞𝓍2 + 𝔟𝓍 + c )6
Answer:
Formulae:
•
•
•
•
Let,
y = (ax2 + bx + c)6
and u = (ax2 + bx + c)
therefore, y = u6
Differentiating above equation w.r.t. x,
………… By chain rule
………….
= 6. (ax2 + bx + c)5. (2ax+b+0) ………….
Question 13.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
•
•
•
Let,
Let, u = (x2-3x+5)3
Therefore, 
For u = (x2-3x+5)3
Let, v = (x2-3x+5)
Therefore, u = (v)3
Therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
…………. 
Question 14.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
•
•
• 
Let,
and 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
…………. 
Question 15.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 1 – sin2x = cos2x
• 
• 
Let,
Multiplying numerator and denominator by (1+sin x),
……………. (1 – sin2x = cos2x)
y = sec x + tan x
Differentiating above equation w.r.t. x,
…………. 
Question 16.
Differentiate each of the following w.r.t. x:
cos2𝓍3
Answer:
Formulae:
• 
•
• 2 sin x. cos x = sin 2x
Let,
y = cos2 x3
and u = x3
therefore, y= cos2 u
let, v = cos u
therefore, y= v2
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
…………. 
Question 17.
Differentiate each of the following w.r.t. x:
sec3 (𝓍2+1)
Answer:
Formulae:
•
• 
Let,
y = sec3 (x2+1)
and u = x2+1
therefore, y= sec3 u
let, v = sec u
therefore, y= v3
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 18.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
• 
Let,
and u = 3x
therefore, 
let, v = cos u
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 19.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
•
•
Let,
and u = 2x
therefore, 
let, v = sin u
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 20.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
•
•
Let,
and u = 1+ cot x
therefore,
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 21.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
Let,
and 
therefore, y= cosec3 u
let, v = cosec u
therefore, y= v3
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 22.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
•
•
Let,
and u = x3
therefore, 
let, v = sin u
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 23.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
•
•
• 
Let,
and u = x. sin x
therefore,
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
…………. 
Question 24.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
•
•
Let,
And
therefore, 
let, v = cot u
therefore,
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 25.
Differentiate each of the following w.r.t. x:
cot3𝓍2
Answer:
Formulae:
•
•
Let,
y = cot3 x2
and u = x2
therefore, y= cot3 u
let, v = cot u
therefore, y= v3
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 26.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
•
•
Let,
and u = ax + b
therefore, 
let, 
therefore, 
let, 
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 27.
Differentiate each of the following w.r.t. x:
Answer:
Formulae:
• 
•
•
•
Let,
and u = x3 + 1
therefore, 
let, 
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
…………. 
Question 28.
Differentiate each of the following w.r.t. x:
sin 5𝓍 cos 3𝓍
Answer:
Formulae:
• 
•
•
•
Let,
y = sin 5x. cos 3x
…………. 
Differentiating above equation w.r.t. x,
…………. 
…………. 
Question 29.
Differentiate each of the following w.r.t. x:
sin 2𝓍 sin 𝓍
Answer:
Formulae:
• 
•
•
• 
Let,
y = sin 2x. sin x
…………. 
Differentiating above equation w.r.t. x,
…………. 
…………. 
Question 30.
Differentiate each of the following w.r.t. x:
cos 4𝓍 cos 2𝓍
Answer:
Formulae:
• 
•
•
•
Let,
y = cos 4x. cos 2x
…………. 
Differentiating above equation w.r.t. x,
…………. 
………….
= -3 sin 6x – sin 2x
= - (3 sin 6x + sin 2x)
Question 31.
Find 
, when:
𝒴 =sin 
Answer:
Formulae:
• 
• 
•
• 
• 
Given,
Put x = tan a
Therefore, 
…………… eq (1)
y = sin (cos 2a) …………. 
Differentiating above equation w.r.t. a ,
…………. 
…………. 
…………. 
But, x = tan a
…………………….. eq (2)
Now,
………… By chain rule
…………….. from eq (1) & eq (2)
…..……. 
…..……. 
Question 32.
Find , when:
𝒴 = 
Answer:
Formulae:
•
•
• 
•
Given,
Differentiating above equation w.r.t. x ,
…………. 
…………. 
Question 33.
If 𝒴 = 
, prove that + 𝒴2 +1 =0.
Answer:
Formulae:
• 
• 
•
•
•
•
• 
Given,
Dividing numerator and denominator by cosx,
…………. 
…………. 
Differentiating above equation w.r.t. x ,
…………. 
………….
…………. 
Now,
…………. 
= 0
Hence Proved.
Question 34.
If 𝒴 = 
, prove that =sec2
.
Answer:
Formulae:
•
• 
•
•
•
•
Given,
Dividing numerator and denominator by cosx,
………….
…………. 
Differentiating above equation w.r.t. x ,
………….
………….
………….
Hence Proved.
Exercise 10j
Question 1.Find the second derivate of :
(i) x11 (ii) 5x
(iii) tan x (iv) cos-1x
Answer:
(i) x11
Differentiating with respect to x
f’(x) = 11x11-1
f’(x)=11x10
Differentiating with respect to x
f’’(x) = 110x10-1
f’’(x)= 110x9
(ii) 5x
Differentiating with respect to x
f’(x)= 5x loge 5 [ Formula: ax = ax logea ]
Differentiating with respect to x
f’’(x)= loge5 . 5x loge5
= 5x(loge5)2
(iii) tan x
Differentiating with respect to x
f’(x)= sec2x
Differentiating with respect to x
f’’(x)= 2secx . secx tanx
= 2sec2x tanx
(iv) cos-1x
Differentiating with respect to x
Differentiating with respect to x
= 
Question 2.
Find the second derivative of:
(i) x sin x
(ii) e2x cos 3x
(iii) x3 log x
Answer:
Differentiating with respect to x
f’(x)= sinx + xcosx
Differentiating with respect to x
f’’(x) = cosx +cosx – xsinx
= -sinx + 2cosx
(ii) e2x cos 3x
Differentiating with respect to x
f’(x) = 2e2xcos3x + e2x(-sin3x).3
= 2e2xcos3x – 3e2xsin3x
Differentiating with respect to x
f’’(x) = 2.2e2xcos3x + 2e2x(-sin3x).3 – 3.2e2xsin3x – 3e2xcos3x.3
= 4e2xcos3x - 6e2xsin3x – 6e2xsin3x – 9e2xcos3x
=-12e2xsin3x – 5e2xcos3x
(iii) x3 log x
Differentiating with respect to x
f’(x) = 
f’(x) = 3x2log x + x2
Differentiating with respect to x
= 6x log x + 3x +2x
= 6x log x +5x
Question 3.
If 𝒴 = x + tan x, show that cos2 . 
- 2𝒴+ 2x = 0.
Answer:
y = x + tan x, ⇒ tan x = y-x…. (i)
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
[putting value of tan x from (i) ]
⇒ 
⇒ 
Question 4.
If 𝒴 = 2 sin x + 3 cos x, s how that 𝒴 + =0.
Answer:
Differentiating with respect to x
Differentiating with respect to x
Hence Proved
Question 5.
If 𝒴 = 3 cos (log x) + 4 sin (log x), prove that x2𝒴2 + x𝒴1 + 𝒴 = 0.
Answer:
Differentiating with respect to x
⇒ 
[ we can also write this as xy1 = -3sin(log x ) +4cos(log x )
Differentiating with respect to x
⇒ 
⇒ 
⇒ x2y2 + xy1 + y = 0
Hence Proved
Question 6.
If y = e- cos x , show that = 2e-xsin x .
Answer:
Differentiating with respect to x
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence proved
Question 7.
If 𝒴 = sec x – tan x. show that (cos x) = 𝒴2.
Answer:
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
⇒ 
Hence Proved
Question 8.
If 𝒴 = (cosec x + cot x), prove that (sin x) - 𝒴2 = 0.
Answer:
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
⇒ 
Hence proved
Question 9.
If 𝒴 = tan-1 , show that (1 + x2) + 2x
=0.
Answer:
Differentiating with respect to x
⇒ 
Differentiating with respect to x
Hence Proved
Question 10.
If 𝒴 = sin (sin x), prove that + (tan ) + 𝒴 cos2x = 0.
Answer:
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 11.
If 𝒴 = a cos (log x) + b sin (log x), prove that x2𝒴2 + x𝒴1 + 𝒴 = 0.
Answer:
Differentiating with respect to x
[ can also be written as -xy1= a sin (log x) ]
Differentiating with respect to x
⇒
⇒ 
Hence Proved
Question 12.
Find the second derivative of e3x sin 4x .
Answer:
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Question 13.
Find the second derivative of sin 3 x cos 5x.
Answer:
Differentiating with respect to x
⇒ 
Differentiating with respect to x
Hence Proved
Question 14.
If 𝒴 = etan , prove that (cos2x) - (1+ sin 2x). = 0.
Answer:
Differentiating with respect to x
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 15.
If 𝒴 = 
, show that 
= 
.
Answer:
Differentiating with respect to x
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence proved
Question 16.
If 𝒴 = eax cos bx, show that - 2a + (a2 + b2) = 0.
Answer:
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Hence Proved
Question 17.
If 𝒴 = ea cos-1x, - 1 ≤ x ≤ 1, show that (1 - x2) - x- a2𝒴 = 0.
Answer:
Taking log on both sides
log y = acos-1x log e
log y = acos-1x
Differentiating with respect to x
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 18.
If x = at2 and 𝒴 = 2at, find at t = 2.
Answer:
Differentiating with t
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Question 19.
If x = a(θ – sin θ ) and 𝒴 = a(1 – cos θ), find at θ = 
.
Answer:
Differentiating with respect to θ
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Question 20.
If 𝒴 = sin (log x), prove that x2+x+𝒴 = 0.
Answer:
Differentiating with respect to
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 21.
If 𝒴 = 
, show that (1 - x2) - 3x- 𝒴 = 0.
Answer:
Differentiating with respect to x
⇒ 
Differentiating with respect to x
⇒ 
Hence Proved
Question 22.
If 𝒴 = ex sin x, prove that - 2 + 2𝒴 = 0.
Answer:
Differentiating with respect to x
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
⇒ 
Question 23.
If x = 
and 𝒴 = a sin θ, show that the value of at θ = 
is 
.
Answer:
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Question 24.
If x = cos t + log tan 
, 𝒴 = sin t then find the values of 
and at t = .
Answer:
Differentiating with respect to t
⇒ 
[Putting t = π /4 ]
⇒ 
⇒ 
⇒ 
Differentiating with respect to x
[Putting t = π /4 ]
⇒ 
⇒ 
⇒ 
Question 25.
If 𝒴 = xx, prove that - 
=0.
Answer:
y = xx
Taking log on both sides
log y = x log x
Differentiating with respect to x
…(i)
⇒ 
Differentiating with respect to x
[putting value of (1 + log x) from (i) ]
⇒ 
⇒ 
Hence Proved
Question 26.
If 𝒴 = (cot-1 )2, then show that (x2 + 1)2+ 2x (x2 + 1) =2.
Answer:
Differentiating with respect to x
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 27.
If 𝒴 = 
, then show that (x2 + 1) + x- m2𝒴 = 0.
Answer:
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
[ 
]
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 28.
If 𝒴 = log [x + 
, then prove that 
Answer:
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒
Hence Proved
Question 29.
If x = a(cos θ + θ sin θ ) and 𝒴 = a(sin θ - θ cos θ), show that = 
Answer:
Differentiating with respect to θ
⇒ 
⇒ 
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Question 30.
If x = a cos θ + b sin θ and 𝒴 = a sin θ – b cos θ , show that 
+ 𝒴 = 0.
Answer:
⇒ 
⇒ 
Differentiating with respect to x
⇒ 
⇒ 
⇒ 
Hence Proved
Exercise 10b
Question 1.Differentiate each of the following w.r.t. x:
(i) 
(ii) 
(iii) 
Answer:
(i) Let y = e4x z = 4x
Formula : 
According to chain rule of differentiation
(ii) Let y = e-5x z = -5x
Formula :
According to chain rule of differentiation
(iii) Let y =
z = x3
Formula : 
According to chain rule of differentiation
Question 2.
Differentiate each of the following w.r.t. x:
(i) 
(ii) 
(iii) 
Answer:
(i) Let y =
z = 2/x
Formula : 
According to chain rule of differentiation
(ii) Let y =
z = 
Formula :
According to chain rule of differentiation
(iii) Let y =
z = -2
Formula :
According to chain rule of differentiation
Question 3.
Differentiate each of the following w.r.t. x:
(i) 
(ii) 
(iii) 
Answer:
(i) Let y =
z = 
Formula : 
According to chain rule of differentiation
(ii) Let y =
z = 
Formula : 
According to chain rule of differentiation
(iii) Let y =
z = 
Formula :
According to chain rule of differentiation
Question 4.
Differentiate each of the following w.r.t. x:
(i) tan (log x)
(ii) log (sec x)
(iii) log (sin (x/2))
Answer:
(i) Let y =tan(log x) z = log x
Formula : 
According to chain rule of differentiation
(ii) Let y = log (sec x) z = sec x
Formula : 
According to chain rule of differentiation
(iii) Let y = log (sin (x/2)) z = sin (x/2)
Formula : 
According to chain rule of differentiation
Question 5.
Differentiate each of the following w.r.t. x:
(i)log3 x
(ii) 
(iii) 
Answer:
(i) Let y = 
Formula : 
Therefore y =
According to chain rule of differentiation
(ii) Let y = z = -x
Formula : 
According to chain rule of differentiation
(iii) Let y = z = x
Therefore Y = 
Formula :
According to chain rule of differentiation
Question 6.
Differentiate each of the following w.r.t. x:
(i) 
(ii) log (sin (3x))
(iii) 
Answer:
(i) Let y = 
z = 
Formula : 
According to chain rule of differentiation
(ii) Let y = log (sin (3x)) z = sin (3x)
Formula :
According to chain rule of differentiation
(iii) Let y = 
z = 
Formula :
According to chain rule of differentiation
Question 7.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = and w = log (x)
Formula : 
According to product rule of differentiation
Question 8.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = 
Formula : 
According to chain rule of differentiation
= 
Question 9.
Differentiate each of the following w.r.t. x:
e2x sin 3x
Answer:
Let y = e2x sin 3x, z = e2x and w = sin 3x
Formula : 
According to product rule of differentiation
Question 10.
Differentiate each of the following w.r.t. x:
e3x cos 2x
Answer:
Let y = e3x cos 2x, z = e3x and w = cos 2x
Formula : 
According to product rule of differentiation
Question 11.
Differentiate each of the following w.r.t. x:
e-5x cot 4x
Answer:
Let y = e-5x cot 4x , z = e-5x and w = cot 4x
Formula : 
According to product rule of differentiation
Question 12.
Differentiate each of the following w.r.t. x:
ex log (sin 2x)
Answer:
Let y = ex log (sin 2x), z = ex and w = log (sin 2x)
Formula : 
According to product rule of differentiation
Question 13.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z =
Formula :
According to chain rule of differentiation
= cosec x
Question 14.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z =
Formula :
According to chain rule of differentiation
= 
Question 15.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, u =
, v =
, z=
Formula :
According to quotient rule of differentiation
If z = 
According to chain rule of differentiation
= 
= 
= 
= 
= 
= 
Question 16.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, u = 
, v = 
Formula :
According to quotient rule of differentiation
If y =
(
)
Question 17.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = x and w =
Formula :
According to product rule of differentiation
Question 18.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = 
and w = 
Formula : 
According to product rule of differentiation
= 
Question 19.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = 
and w = 
Formula : 
According to product rule of differentiation
Question 20.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, u = 
, v =
Formula: 
According to quotient rule of differentiation
If y =
Question 21.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = x3 and w = 
Formula :
According to product rule of differentiation
= 
Question 22.
Differentiate each of the following w.r.t. x:
Answer:
Let y = 
, z = xcos x
Formula :
(Using product rule)
According to chain rule of differentiation
= 
Exercise 10c
Question 1.Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and u = 2x
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 2.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and u = x2
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 3.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and 
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 4.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii)
Answer :
Let,
and 
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 5.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and u = log x
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 6.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and u = ex
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 7.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii)
Answer :
Let,
and u = tan-1x
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 8.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
Answer :
Let,
and u = x3
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 9.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii) 
iii) 
Answer :
Let,
and 
therefore,
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
………
Question 10.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i) 
ii) 
iii)
iv) 
v)
Answer :
Let,
Let, u = (1+x2) and v=tan-1x
therefore, y=u.v
………
…………. 
…………. 
Question 11.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii) 
iii) 
Answer :
Let,
and u = cot x
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
………
= -1
Question 12.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii)
iii)
Answer :
Let,
and u = x4
therefore, 
let, 
therefore, y= log v
Differentiating above equation w.r.t. x,
………… By chain rule
…………. 
Question 13.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii)
Answer :
Let,
and u = x2
therefore, 
let, 
therefore, y= v3
Differentiating above equation w.r.t. x,
………… By chain rule
………….
Question 14.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii)
iii) 
Answer :
Let,
and
therefore, 
let, 
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
……… 
Question 15.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii) 
Answer :
Let,
and u = sin-1x
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
Question 16.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii)
iii) 
Answer :
Let,
and
therefore, 
let, 
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
……… 
Question 17.
Differentiate each of the following w.r.t. x:
Answer:
Formulae :
i)
ii)
iii) 
Answer :
Let,
and 
therefore, 
let,
therefore, 
Differentiating above equation w.r.t. x,
………… By chain rule
……… 
Question 18.
If 
prove that 
.
Answer:
Given : 
To Prove : 
Formulae :
i)
ii)
iii)
iv) 
v)
vi) 
Answer :
Given equation,
Let 
Therefore, y = s + t ………eq(1)
I. For
let
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
………….
………
………eq(2)
II. For
let 
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
…………. 
………
………eq(2)
Differentiating eq(1) w.r.t. x,
………
= -1 -1 ………from eq(2) and eq(3)
= -2
Hence proved !!!
Question 19.
Prove that 
.
Answer:
To Prove:
Formulae :
i) 
ii)
iii) 
iv) 
v) 
vi)
vii) 
Answer :
Let,
Let 
Therefore, y = s - t ………eq(1)
I. For
let 
therefore, 
Differentiating above equation w.r.t. x,
………
…………. 
………eq(2)
II. For
let 
therefore, 
Differentiating above equation w.r.t. x,
…………. By chain rule
………
……… 
………eq(3)
Differentiating eq(1) w.r.t. x,
………
………from eq(2) and eq(3)
= 2 tan-1x
Hence proved !!!
Exercise 10d
Question 1.Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 
We have,
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that 
Now differentiating 
Question 2.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 
(ii) 1 + cos 
= 2
We have,
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that 
Now differentiating
Question 3.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i)
(ii) 1 + cos = 2
We have,
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that 
Now differentiating
Question 4.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i)
(ii) 1 + cos = 2
We have,
⇒ 
⇒
⇒
Now, we can see that 
Now differentiating
Question 5.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 
We have, 
Dividing numerator and denominator by cosx
Now, we can see that 
Now differentiating
⇒ 0 + 1
⇒ 1
Ans) 1
Question 6.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 
We have,
Dividing numerator and denominator by cosx
Now, we can see that 
Now differentiating
⇒ 0 + 1
⇒ 1
Ans) 1
Question 7.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 1 - cos = 2
(ii) 1 + cos = 2
We have,
Now, we can see that 
Now differentiating
Question 8.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) cos 
= 
We have,
Dividing numerator and denominator by 1+tan2x
⇒ 2x
Now, we can see that 
Now differentiating
⇒ 2
Ans) 2
Question 9.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) cos =
We have,
Now, we can see that 
Now differentiating
⇒ 0 - 2
⇒ -2
Ans) -2
Question 10.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) sin = 
We have,
Dividing Numerator and Denominator with 1+tan2x
⇒ 2x
Now, we can see that 
Now differentiating
⇒ 2
Ans) 2
Question 11.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
Formula used: (i) 
(ii) 1 + cos = 2
We have,
⇒
⇒
⇒
⇒
Now, we can see that 
Now differentiating
Question 12.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i) 
(ii) 
We have,
⇒ 
Now, we can see that 
Now differentiating
⇒ -2
Ans) -2
Question 13.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i) 
(ii) 
We have,
⇒ Putting x = cosθ
θ = cos-1x … (i)
Putting x = cosθ in the equation
⇒ 
⇒ 
⇒ 
⇒
[From (i)]
Ans) 
Question 14.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = cosθ
θ = cos-1x … (i)
Putting x = cosθ in the equation
⇒ 
⇒ 
⇒ 
Now, we can see that = 
⇒ θ = cos-1x
Ans) 
Question 15.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = cosθ
θ = cos-1x … (i)
Putting x = cosθ in the equation
⇒ 
⇒ 
⇒ 
Now, we can see that =
⇒ θ = cos-1x
Ans)
Question 16.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
Now, we can see that = θ
⇒ θ = sin-1x
Ans) 
Question 17.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 2θ
⇒ 2sin-1x
Now, we can see that 
= 2sin-1x
Now Differentiating
Ans) 
Question 18.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 3θ
Now, we can see that = 3θ
Now Differentiating
Ans) 
Question 19.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i) 
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 20.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ θ
Now, we can see that = θ
Now Differentiating
Question 21.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ θ
Now, we can see that = θ
Now Differentiating
Question 22.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 23.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have, 
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒
Now, we can see that =
Now Differentiating
Question 24.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = sinθ
θ = sin-1x … (i)
Putting x = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 2
Now, we can see that =
Now Differentiating
Question 25.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = cotθ
θ = cot-1x … (i)
Putting x = cotθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒
⇒ θ
Now, we can see that =
Now Differentiating
Question 26.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 27.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 28.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒
⇒ 
Now, we can see that =
Now Differentiating
Question 29.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒
Now, we can see that =
Now Differentiating
Question 30.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒
⇒
⇒
Now, we can see that =
Now Differentiating
Question 31.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 32.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = tanθ
θ = tan-1x … (i)
Putting x = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that =
Now Differentiating
Question 33.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ 
⇒ Putting xn = tanθ
θ = tan-1
… (i)
Putting xn = tanθ in the equation
⇒ 
⇒ 
⇒
⇒ 
Now, we can see that =
Now Differentiating
Question 34.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x = asinθ
sinθ = 
θ = 
… (i)
Putting x = asinθ in the equation
⇒ 
⇒ 
⇒ 
⇒
⇒
Now, we can see that =
Now Differentiating
Question 35.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting ax = sinθ
θ = 
… (i)
Putting ax = sinθ in the equation
⇒ 
⇒ 
⇒ 
⇒
⇒ 
Now, we can see that =
Now Differentiating
Question 36.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting ax = tanθ
θ = 
… (i)
Putting ax = tanθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that 
=
Now Differentiating
Question 37.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ Putting x2 = a2 cotθ
θ = 
… (i)
Putting x2 = a2 cotθ in the equation
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
Now, we can see that 
=
Now Differentiating
Question 38.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
⇒ 
Putting e2x = tanθ
θ = 
… (i)
Putting e2x = tanθ in the equation
Now, we can see that 
=
Now Differentiating
Question 39.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i) 
(ii) 
We have,
Putting 2x = cosθ
θ = 
… (i)
Putting e2x = tanθ in the equation
⇒ 
Now, we can see that 
=
Now Differentiating
Question 40.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii) 
We have,
⇒ tan-1a – tan-1x
Now Differentiating
Question 41.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
Now Differentiating
Question 42.
Differentiate each of the following w.r.t x:
Answer:
To find: Value of 
The formula used: (i)
(ii)
We have,
Now Differentiating
Question 43.
Differentiate each of the following w.r.t x:
Answer:
Given: Value of 
The formula used: (i)
(ii)
We have,
Now Differentiating
Question 44.
Differentiate each of the following w.r.t x:
Answer:
Given: Value of 
The formula used: (i)
(ii)
We have,
Now Differentiating
Question 45.
Differentiate each of the following w.r.t x:
Answer:
Given: Value of 
The formula used: (i)
(ii)
We have,
Now Differentiating
Question 46.
Differentiate each of the following w.r.t x:
If 
, prove that 
.
Answer:
Given: Value of 
To Prove: 
The formula used: (i)
(ii)
We have,
Dividing numerator and denominator with a
Now Differentiating
Question 47.
Differentiate each of the following w.r.t x:
If 
, show that 
.
Answer:
Given: Value of 
To Prove: 
The formula used: (i)
(ii)
We have, 
Putting x = tanθ
θ = tan-1x
Dividing numerator and denominator with a
⇒ 4tan-1x
Now Differentiating
Question 48.
Differentiate each of the following w.r.t x:
If 
, show that 
Answer:
Given: Value of 
To Prove: 
Formula used: (i)
(ii)
We have, 
⇒ 
Now Differentiating
Question 49.
Differentiate each of the following w.r.t x:
If 
, show that 
.
Answer:
Given: Value of 
To Prove: 
Formula used: (i)
Let x = cosθ
θ = cos-1x
Putting x = cosθ in equation
Now Differentiating
Question 50.
Differentiate each of the following w.r.t x:
If 
. Prove that 
.
Answer:
Given: Value of 
To Prove: 
The formula used: (i)
(ii)
Let x = cos2θ
2θ = cos-1x
θ = 
cos-1x
Putting x = cos2θ
Dividing by cosθ in the numerator and denominator
Now Differentiating
Question 51.
Differentiate each of the following w.r.t x:
Differentiate 
w. r. t. x
Answer:
Given: Value of 
To find: 
The formula used: (i)
(ii)
Let 2x = tanθ
θ = tan-1(2x)
Putting 2x = tanθ
Now Differentiating
Exercise 10e
Question 1.Find , when:
x2 + y2 = 4
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to the chain rule of differentiation
Therefore ,
Question 2.
Find , when:
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to the chain rule of differentiation
Therefore ,
Question 3.
Find , when:
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to the chain rule of differentiation
Therefore ,
Question 4.
Find , when:
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to the chain rule of differentiation
Therefore ,
Question 5.
Find , when:
xy = c2
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to product rule of differentiation
Therefore ,
Question 6.
Find , when:
x2 + y2 — 3xy = 1
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 7.
Find , when:
xy2 — x2y — 5 = 0
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 8.
Find , when:
(x2 + y2)2 = xy
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 9.
Find , when:
x2 + y2 = log (xy)
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 10.
Find , when:
xn + yn = an
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to the chain rule of differentiation
Therefore ,
Question 11.
Find , when:
x sin 2y = y cos 2x
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to the chain rule of differentiation
According to the product rule of differentiation
Therefore ,
Question 12.
Find , when:
sin2x + 2cos y + xy
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to chain rule of differentiation
Therefore ,
Question 13.
Find , when:
y sec x + tan x + x2y = 0
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to product rule of differentiation
Therefore ,
Question 14.
Find , when:
cot (xy) + xy = y
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
(Since, 
)
Question 15.
Find , when:
y tan x — y2 cos x + 2x = 0
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 16.
Find , when:
ex log y = sin—1 x + sin —1y
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
Question 17.
Find , when:
xy log (x + y) = 1
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to product rule of differentiation
Therefore ,
(Multiply and divide by x)
Question 18.
Find , when:
tan (x + y) + tan (x — y) = 1
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
Therefore ,
Question 19.
Find , when:
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to quotient rule of differentiation
Therefore ,
Question 20.
Find , when:
If y = x sin y , prove that 
.
There is correction in question …. Prove that should be
instead of to get the required answer.
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
Therefore ,
Question 21.
Find , when:
If xy = tan (xy), show that 
.
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to product rule of differentiation
Therefore ,
Question 22.
Find , when:
If y log x = (x — y), prove that 
.
Answer:
Let us differentiate the whole equation w.r.t x
Formula :
According to product rule of differentiation
Therefore ,
(Multiply by 1+log x on both sides)
(y log x = x - y)
Question 23.
Find , when:
If cos y = x cos (y + a), prove that 
.
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to chain rule of differentiation
According to product rule of differentiation
Therefore ,
( Multiply and divide by cos (y+a) )
(Since cos y = x cos (y + a))
(Formula sin(a-b)=sin a cos b – cos a sin b)
Question 24.
Find , when:
If 
, prove that 
.
Answer:
Let us differentiate the whole equation w.r.t x
Formula : 
According to the chain rule of differentiation
Therefore ,
Exercise 10f
Question 1.Find 
, when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x we get,
Question 2.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 3.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 4.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 5.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 6.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 7.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 8.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 9.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 10.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 11.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 12.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 13.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 14.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 15.
Find , when:
Answer:
Here, the argument of the sinusoidal function has exponent as x itself.
For that, we will consider 
for simplicity.
Differentiating both the sides,
. ……..(1)
Now we have to find 
, where 
take log both the sides
Now differentiating both sides by x, we get,
Substituting the value in equation 1,
Question 16.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 17.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 18.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 19.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 20.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 21.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 22.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 23.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 24.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 25.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 26.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 27.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 28.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 29.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 30.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 31.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 32.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 33.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 34.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 35.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
for v we do not have to take log just simply differentiate it,
……..(2)
Question 36.
Find , when:
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Here there are three terms to differentiate for this; we can take two term as one and then apply product rule, I am taking 
as a single term
Differentiate
……….(1)
for v we do not have to take log just simply differentiate it,
……..(2)
Question 37.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 38.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 39.
Find , when:
Answer:
Here we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 40.
Find , when:
Answer:
Here, we need to take log both the sides to get that differentiation simple.
Now differentiating both sides by x, we get,
Question 41.
Find , when:
Answer:
. So the equation given is implicit, we will just take log both sides
Now differentiate it with respect to x and consider 
Taking y’ one side, we get
Question 42.
Find , when:
Answer:
. So the equation given is implicit, we will just take log both sides
Now differentiate it with respect to x and consider
Taking y’ one side, we get
Question 43.
Find , when:
Answer:
we can write this equation as,
Differentiate
Question 44.
If 
, prove that 
.
Answer:
differentiate the given y to get the result,
Question 45.
If 
, prove that 
.
Answer:
differentiate the given y to get the result,
{taking y’ one side}
Question 46.
If 
, prove that 
.
Answer:
taking log both sides,
differentiating both sides,
Take y’ one side,
Question 47.
If 
, show that 
Answer:
differentiate both sides,
Taking y’ one side,
Question 48.
If 
, prove that 
Answer:
differentiate both sides,
Question 49.
If 
, prove that 
.
Answer:
differentiate both sides,
Question 50.
If 
, prove that 
.
Answer:
differentiate both sides,
Question 51.
If 
, show that 
Answer:
differentiate both sides,
Question 52.
If 
, show that 
Answer:
differentiate both sides,
Question 53.
If 
, show that 
.
Answer:
differentiate both sides,
Question 54.
If 
, show that 
.
Answer:
differentiate both sides,
Question 55.
If 
, prove that 
.
.
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 56.
If 
, prove tha
.
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 57.
If 
, 
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 58.
If 
, prove that 
.
Answer:
simply taking log both sides would not help more.
For that let us assume and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 59.
If 
prove that 
.
Answer:
simply taking log both sides would not help more.
For that let us assume 
and 
Take log both sides
Differentiate
……….(1)
Take log both sides,
Differentiate ,
……(2)
Question 60.
If 
, find .
Answer:
equality is not given but we may assume that it is equal to 0.
We can also write this equation as
Now differentiating it,
Question 61.
If 
, find 
.
Answer:
take log both the side,
Now differentiate it,
To get f’(0) we need to find f(0),
Putting x=0 in f
Now put x=0 in f’(x),
Question 62.
If 
find .
Answer:
we can write this equation as,
Differentiate it,
Question 63.
If 
, find .
Answer:
simply differentiate both sides,
Take y’ one side
Question 64.
, find .
Answer:
we can write this as,
Differentiate ,
Question 65.
Find , when:
If 
, prove that 
.
Answer:
Differentiate it,
Question 66.
If 
, prove that 
.
Answer:
taking log both sides,
differentiating both sides,
Take y’ one side,
Exercise 10g
Question 1.If 
, prove that
Answer:
Given : 
To prove : 
Formula used : 
= 
= 
= cos x
If u and v are functions of x,then 
= u
+ v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = 
taking log on both sides
log y = log
log y = y log (
)
Differentiating both sides with respect to x
= 
= 
log() + y 
= log() + y 
= 
log() + y 
log sinx) = y
= y
Question 2.
If 
, prove that 
Answer:
Given : 
To prove : 
Formula used : =
=
= - sinx
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Given that y = 
taking log on both sides
log y = log 
log y = y log (
)
Differentiating both sides with respect to x
= 
= log() + y 
= log() + y 
= log() + y 
log) = - y
Question 3.
If 
, prove that 
Answer:
Given : 
To prove : 
Formula used : =
=
= 1
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
.
y = 
squaring on both sides
= x + y
Differentiating with respect to x
2y = 1 +
(2y – 1) = 1
= 
=
Question 4.
If
, prove that
Answer:
Given : 
To prove : 
Formula used : =
=
= - sinx
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = 
squaring on both sides
= 
+ y
Differentiating with respect to x
2y = 
+
(2y – 1) =
= 
= 
=
=
Question 5.
If 
prove that 
Answer:
Given : 
To prove : 
Formula used : =
=
= 
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = 
squaring on both sides
= + y
Differentiating with respect to x
2y = +
(2y – 1) = 
= 
= 
=
=
Question 6.
If 
, show that 
Answer:
Given : 
To show : 
Formula used : =
=
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = 
squaring on both sides
= 
+ y
Differentiating with respect to x
2y = 
+
(2y – 1) =
(2y – 1) =
Question 7.
If 
, prove that 
Answer:
Given : 
To show : 
Formula used : =
=
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = 
taking log on both sides
log y = log
log y = 
log a
taking log on both sides
log(log y) = log(
)
log(log y) = y.logx + log()
Differentiating both sides with respect to x
= 
+ 0 (as differentiation of log() [constant] is zero )
= 
+ y.
= + y.
= 
(
=
.
Question 8.
If 
prove that 
Answer:
Given :
To show : 
Formula used : =
=
If u and v are functions of x,then = u + v
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
y = x + 
y2 = xy + 1
Differentiating with respect to x
= 
+ 0 (as differentiation of constant is zero )
2y. = x. + y
= y
= 
=
Exercise 10h
Question 1.Differentiate 
with respect to 
Answer:
Given : Let u = x6 and v = 
To differentiate : x6 with respect to
Formula used : 
n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = x6 and v =
Differentiating u with respect to x
= 6x5
Differentiating v with respect to x
= 
= 
= 
= -12
= -12
Ans. 
Question 2.
Differentiate log 
with respect to 
Answer:
Given : Let u = log x and v = cot x
To differentiate : log xwith respect to cot x
Formula used : 
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = log x and v = cot x
Differentiating u with respect to x
=
Differentiating v with respect to x
= 
=
= 
= 
Question 3.
Differentiate 
with respect to 
.
Answer:
Given : Let u = 
and v = cos x
To differentiate : with respect to cos x
Formula used : 
= - sinx
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = 
and v = cos x
Differentiating u with respect to x
= 
= .
Differentiating v with respect to x
= 
=
= 
= 
.
Ans. 
Question 4.
Differentiate 
with respect to 
Answer:
Given : Let u = and v =
To differentiate : 
with respect to
Formula used : 
n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
= 
Let u = and v =
Differentiating u with respect to x
= 
= 
. 
= 
.
= 
.
= 
= 
= 
.
= 
= 
=
Differentiating v with respect to x
= 
.
= 
=
=
= 
= 
=
Ans. 
Question 5.
Differentiate 
with respect to 
Answer:
Given : Let u = and v =
To differentiate : 
with respect to
Formula used : n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
=
Let u = 
and v = 
Differentiating u with respect to x
= 
= 
. 
= 
= 
. 
= 
. 
= 
= 
=
Differentiating v with respect to x
= 
. 
= 
.
= 
.
= 
.
= 
. 
= 
=
=
= 
= 1
= 1
Ans. 1
Question 6.
Differentiate with respect to 
Answer:
Given : Let u = and v =
To differentiate : 
with respect to 
Formula used : n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
=
Let u = and v =
Differentiating u with respect to x
= 
= 
. 
= 
= 
. 
= 
) . 
= 
=
Differentiating v with respect to x
= 
= 
. 
= 
4x
= 
= 
= 
=
=
= 
= 
=
Ans.
Question 7.
Differentiate 
with respect to 
Answer:
Given : Let u = 
and v = 
To differentiate : with respect to
Formula used : n.
= cos x
= - sin x
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = and v =
Differentiating u with respect to x
= 
.
= 
=
Differentiating v with respect to x
= 
. = 
= 
=
= 
= 
= 
=
Ans. 
Question 8.
Differentiate 
with respect to 
Answer:
Given : Let u = and v =
To differentiate : with respect to
Formula used : 
= 
n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
=
Let u = and v =
Differentiating u with respect to x
= 
= 
. 
= 
= 
. 
= 
. 
= 
=
For v =
Let x = tan θ
= 
= 
) = 3θ = 3
= 3
Differentiating v with respect to x ,
= 
= 
=
=
= 
= 
=
Ans. 
Question 9.
Differentiate 
with respect to 
Answer:
Given : Let u = and v =
To differentiate : with respect to
Formula used : n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = and v =
Put x = cot θ or θ = 
in u
= 
= 
= 
= 
= 
= 
We know that 1 - 
= 
- 2
and 
= 
1 - = 
Substituting the above values in ,we get
= 
= 
= 
Dividing by cos 
on numerator and denominator,we get
= 
= 
=
= 
= 
Differentiating u with respect to x
= 
= 
=
v = 
Put x = tanθ
V = = 
= 
= 
= 
V = = 
= 
= 2θ = 2
V = = 2
Differentiating v with respect to x
= 
=
= 
= 
=
Ans. 
Question 10.
Differentiate 
with respect to 
when 
Answer:
Given : Let u = and v =
To differentiate : with respect to
Formula used : n.
The CHAIN RULE states that the derivative of f(g(x)) is f’(g(x)).g’(x)
Let u = and v =
Substitute x = cosθ in u
u = 
= 
= 
u = 
= 
= 
u = = 
Differentiating u with respect to x
= 
Substitute x = sinθ in v ,
v = 
= 
= 
v = = 
= 
v = = 
= 
v = = 
v = 
Differentiating v with respect to x
= 
=
= 
=
Ans. 
Exercise 10i
Question 1.Find , when
𝒳 = at2, 𝒴 = 2at
Answer:
Theorem: y and x are given in a different variable that is t. We can find 
by finding 
and 
and then dividing them to get the required thing.
= 
= 2a. ……(1)
= 
= 2at ……(2)
Dividing (1) and (2), we get
= 
= 
Question 2.
Find , when
𝒳 = a cos θ , 𝒴 = b sin θ
Answer:
y and x are given in a different variable that is θ . We can find by finding 
and 
and then dividing them to get the required thing.
= 
(
=cosθ )
= bcosθ. ……(1)
(
)
= -asinθ …….(2)
Dividing (1) and (2), we get
= 
(
= cotθ )
= 
.
Question 3.
Find , when
𝒳 = a cos2θ , 𝒴 = b sin2θ
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= b× 2sinθ × cosθ (using the chain rule 
= 2sinθ× = 2sinθ × cosθ )
= 2bsinθcosθ . ………..(1)
= a × (2cosθ)× (-sinθ ) (using chain rule 
= 2cosθ× 
= 2 cosθ × (-sinθ ) )
= -2asinθcosθ.
Dividing (1) and (2), we get
= 
= 
.
Question 4.
Find , when
𝒳 = a cos3θ, 𝒴 = a sin3θ
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= a× 3 sin2θ × cosθ (using the chain rule 
= 3sin2θ× = 2sin2θ × cosθ )
= 3asin2θcosθ . ………..(1)
= a × (3cos2θ)× (-sinθ ) (using chain rule = 2cosθ× = 2 cosθ × (-sinθ ) )
= -3asinθcos2θ.
Dividing (1) and (2), we get
= 
= 
.
= -tanθ
Question 5.
Find , when
𝒳 = a(1 – cos θ), 𝒴 = a(θ + sin θ)
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= a× (1+cosθ ) ………..(1)
= asinθ. …………….(2)
Dividing (1) and (2), we get
= 
= 
.
= 
( 1+cosθ=2cos2θ/2 and sinθ = 2sin(θ/2)cos(θ/2))
= cot(θ/2)
Question 6.
Find , when
𝒳 = a log t, 𝒴 = b sin t
Answer:
Theorem: y and x are given in a different variable that is t. We can find by finding and and then dividing them to get the required thing.
= 
= bcost ………..(1)
= 
…………(2)
Dividing (1) and (2), we get
= 
= 
.
Question 7.
Find , when
𝒳 = (log t + cos t), 𝒴 = (et + sin t)
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= et + cost ………..(1) 
= - sint. …………….(2) 
Dividing (1) and (2), we get
= 
= 
.
Question 8.
Find , when
𝒳 = cos θ + cos 2θ, 𝒴 = sin θ + sin 2θ
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= cosθ + cos2θ×2 ………..(1) 
= -sinθ -2sin2θ …………….(2) 
Dividing (1) and (2), we get
= 
Question 9.
Find , when
𝒳 = 
, 𝒴 = 
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= 
= 
………..(1)
= 
= 
……..(2)
Dividing (2) and (2), we get
= 
= 
= -( tan2θ)3/2
Question 10.
Find , when
𝒳 = eθ (sin θ + cos θ), 𝒴 = eθ (sin θ - θ cos θ)
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= eθ (cosθ + sinθ ) + (sinθ -cosθ )eθ ………..(1) {by using product rule, 
}
= eθ (cosθ - sinθ ) + eθ (sinθ + cosθ ) …………….(2) {by using product rule, }
Dividing (1) and (2), we get
= 
=tanθ .
Question 11.
Find , when
𝒳 = a (cos θ + θ sin θ ), 𝒴 = a (sin θ - θ cos θ)
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= a(cosθ - (-θ sinθ + cosθ )) {by using product rule, while differentiating θcosθ }
= a(θsinθ ) ………..(1)
= a(-sinθ +θ cosθ +sinθ ) {by using product rule, while differentiating θcosθ }
= a× θcosθ ……….(2)
Dividing (1) and (2), we get
= 
= tanθ ANS
Question 12.
Find , when
𝒳 = 
, 𝒴 = 
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= 
{by using divide rule, 
}
= 
= 
………..(1)
= 
{by using divide rule, 
}
= 
= 
……….(2)
Dividing (1) and (2), we get
= 
= 
Question 13.
Find , when
𝒳 = 
, 𝒴 = 
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= 
{by using divide rule, }
= 
= 
………..(1)
= 
{by using divide rule, }
= 
= 
……….(2)
Dividing (1) and (2), we get
= 
= 
Question 14.
Find , when
𝒳 = cos-1
, 𝒴 = sin-1
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
Let us assume u= 
= 
= 
= 
{by using divide rule, }
Putting value of u
= 
= 
………..(1)
Let assume v= 
= 
{by using divide rule, }
Putting value of v
= 
= 
= 
……….(2)
Dividing (1) and (2), we get
= 
= 
Question 15.
If 𝒳 = 2 cos t – 2 cos3t, 𝒴 = sin t – 2 sin3t, show that =cot t.
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= cost – 6 sin2t × cost ………..(1) 
= -2sint + 6cos2t ×sint …………….(2)
Dividing (1) and (2), we get
= 
= .
Question 16.
If 𝒳 = 
and 𝒴 = 
=t.
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= 
= 
………..(1)
= 
{by using divide rule, }
= 
…………….(2)
Dividing (1) and (2), we get
= 
= t.
Question 17.
If 𝒳 = a(θ – sin θ), 𝒴 = a(1 – cos θ), find at θ =
.
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= asinθ ………..(1)
= a(1-cosθ ) ……….(2)
Dividing (1) and (2), we get
= 
Putting θ= π/2
= 
= 1.
Question 18.
If 𝒳 = 2 cosθ – cos 2θ and 𝒴 = 2 sin θ – sin 2θ, show that =tan 
.
Answer:
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
= 2cosθ – 2cos2θ ………..(1)
= -2sinθ + 2sin2θ ……….(2)
Dividing (1) and (2), we get
= 
= 
= 
By factorising numerator, we get
= 
= 
= 
Foe simplicity let’s take θ/2 as x.
= 
= 
= 
= 
= 
Question 19.
If 𝒳 = 
, 𝒴 = 
, find .
Answer:
Theorem: y and x are given in a different variable that is t . We can find by finding and and then dividing them to get the required thing.
= 
= 
{by using divide rule, }
= 
= 
= 
= 
= 
……..(1)
= 
{by using divide rule, }
= 
= 
= 
= 
= 
……..(1)
Dividing (1) and (2), we get
= 
= 
Question 20.
If 𝒳 = (2 cos θ – cos 2θ) and = (2sin θ – sin 2θ), find 
.
Answer:
here we have to find the double derivative, so to find double derivative we will just differentiate the first derivative once again with a similar method.
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
=
= 2cosθ – 2cos2θ ………..(1)
= -2sinθ + 2sin2θ ……….(2)
Dividing (1) and (2), we get
=
{as shown in question no. 18}
Let 
⇒ To find f’’ we will differentiate f’ with θ and then divide with equation (2).
=
= 
Now divide by equation (2).
Putting θ = π/2
Question 21.
If 𝒳 = a (θ – sin θ), 𝒴 = a( 1+cos θ), find .
Answer:
here we have to find the double derivative, so to find double derivative we will just differentiate the first derivative once again with a similar method.
Theorem: y and x are given in a different variable that is θ . We can find by finding and and then dividing them to get the required thing.
= 
………..(1)
= a(1-cosθ ) ……….(2)
Dividing (1) and (2), we get
= 
= 
= -cot (θ/2)
⇒ To find f’’ we will differentiate f’ with θ and then divide with equation (2).
= 
= 
= 