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RD Sharma solutions for Class 12 Maths chapter 11 - Differentiation [Latest edition]

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Chapter 11: Differentiation

Exercise 11.1Exercise 11.2Exercise 11.3Exercise 11.4Exercise 11.5Exercise 11.6Exercise 11.7Exercise 11.8Others
Exercise 11.1 [Page 17]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.1 [Page 17]

Exercise 11.1 | Q 1 | Page 17

Differentiate the following functions from first principles e−x.

Exercise 11.1 | Q 2 | Page 17

Differentiate the following functions from first principles e3x.

Exercise 11.1 | Q 3 | Page 17

Differentiate the following functions from first principles eax+b.

Exercise 11.1 | Q 4 | Page 17

Differentiate the following functions from first principles ecos x.

Exercise 11.1 | Q 5 | Page 17

Differentiate the following functions from first principles  \[e^\sqrt{2x}\].

Exercise 11.1 | Q 6 | Page 17

Differentiate the following functions from first principles log cos x ?

Exercise 11.1 | Q 7 | Page 17

​Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .

Exercise 11.1 | Q 8 | Page 17

Differentiate the following functions from first principles x2ex ?

Exercise 11.1 | Q 9 | Page 17

Differentiate the following functions from first principles log cosec x ?

Exercise 11.1 | Q 10 | Page 17

Differentiate the following functions from first principles sin−1 (2x + 3) ?

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Exercise 11.2 [Pages 37 - 38]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.2 [Pages 37 - 38]

Exercise 11.2 | Q 1 | Page 37

Differentiate sin (3x + 5) ?

Exercise 11.2 | Q 2 | Page 37

Differentiate tan2 x ?

Exercise 11.2 | Q 3 | Page 37

Differentiate tan (x° + 45°) ?

Exercise 11.2 | Q 4 | Page 37

Differentiate sin (log x) ?

Exercise 11.2 | Q 5 | Page 37

Differentiate \[e^{\sin} \sqrt{x}\] ?

Exercise 11.2 | Q 6 | Page 37

Differentiate etan x ?

Exercise 11.2 | Q 7 | Page 37

Differentiate sin2 (2x + 1) ?

Exercise 11.2 | Q 8 | Page 37

Differentiate log7 (2x − 3) ?

Exercise 11.2 | Q 9 | Page 37

Differentiate tan 5x° ?

Exercise 11.2 | Q 10 | Page 37

Differentiate `2^(x^3)` ?

Exercise 11.2 | Q 11 | Page 37

Differentiate \[3^{e^x}\] ?

Exercise 11.2 | Q 12 | Page 37

Differentiate logx 3 ?

Exercise 11.2 | Q 13 | Page 37

Differentiate \[3^{x^2 + 2x}\] ?

Exercise 11.2 | Q 14 | Page 37

Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?

Exercise 11.2 | Q 15 | Page 37

Differentiate \[3^{x \log x}\] ?

Exercise 11.2 | Q 16 | Page 37

Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?

Exercise 11.2 | Q 17 | Page 37

Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?

Exercise 11.2 | Q 18 | Page 37

Differentiate (log sin x)?

Exercise 11.2 | Q 19 | Page 37

Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?

Exercise 11.2 | Q 20 | Page 37

Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?

Exercise 11.2 | Q 21 | Page 37

Differentiate \[e^{3 x} \cos 2x\] ?

Exercise 11.2 | Q 22 | Page 37

Differentiate sin(log sin x) ?

Exercise 11.2 | Q 23 | Page 37

Differentiate \[e^{\tan 3 x} \] ?

Exercise 11.2 | Q 24 | Page 37

Differentiate \[e^\sqrt{\cot x}\] ?

Exercise 11.2 | Q 25 | Page 37

Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?

Exercise 11.2 | Q 26 | Page 37

Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?

Exercise 11.2 | Q 27 | Page 37

Differentiate \[\tan \left( e^{\sin x }\right)\] ?

Exercise 11.2 | Q 28 | Page 37

Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?

Exercise 11.2 | Q 29 | Page 37

Differentiate \[\frac{e^x \log x}{x^2}\] ? 

Exercise 11.2 | Q 30 | Page 37

Differentiate \[\log \left( cosec x - \cot x \right)\] ?

Exercise 11.2 | Q 31 | Page 37

Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?

Exercise 11.2 | Q 32 | Page 37

Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?

Exercise 11.2 | Q 33 | Page 37

Differentiate \[\tan^{- 1} \left( e^x \right)\] ?

Exercise 11.2 | Q 34 | Page 37

Differentiate \[e^{\sin^{- 1} 2x}\] ?

Exercise 11.2 | Q 35 | Page 37

Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?

Exercise 11.2 | Q 36 | Page 37

Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?

Exercise 11.2 | Q 37 | Page 37

Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?

Exercise 11.2 | Q 38 | Page 37

Differentiate \[\log \left( \tan^{- 1} x \right)\]? 

Exercise 11.2 | Q 39 | Page 37

Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?

Exercise 11.2 | Q 40 | Page 37

Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?

Exercise 11.2 | Q 41 | Page 37

Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?

Exercise 11.2 | Q 42 | Page 37

Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?

Exercise 11.2 | Q 43 | Page 37

Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?

Exercise 11.2 | Q 44 | Page 37

Differentiate  \[e^x \log \sin 2x\] ?

Exercise 11.2 | Q 45 | Page 37

Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?

Exercise 11.2 | Q 46 | Page 37

Differentiate `log [x+2+sqrt(x^2+4x+1)]`

Exercise 11.2 | Q 47 | Page 37

Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?

Exercise 11.2 | Q 48 | Page 37

Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?

Exercise 11.2 | Q 49 | Page 37

Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?

Exercise 11.2 | Q 50 | Page 37

Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?

Exercise 11.2 | Q 51 | Page 37

Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?

Exercise 11.2 | Q 52 | Page 38

Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?

Exercise 11.2 | Q 53 | Page 38

\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?

Exercise 11.2 | Q 54 | Page 38

Differentiate \[e^{ax} \sec x \tan 2x\] ?

Exercise 11.2 | Q 55 | Page 38

Differentiate \[\log \left( \cos x^2 \right)\] ?

Exercise 11.2 | Q 56 | Page 38

Differentiate \[\cos \left( \log x \right)^2\] ?

Exercise 11.2 | Q 57 | Page 38

Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?

Exercise 11.2 | Q 58 | Page 38

If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?

Exercise 11.2 | Q 59 | Page 38

 If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?

Exercise 11.2 | Q 60 | Page 38

If \[y = \frac{x}{x + 2}\]  , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ? 

Exercise 11.2 | Q 61 | Page 38

If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?

 

Exercise 11.2 | Q 62 | Page 38

If  \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\]  prove that \[\frac{dy}{dx} = \sec 2x\] ?

Exercise 11.2 | Q 63 | Page 38

If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that  \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?

Exercise 11.2 | Q 64 | Page 38

If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] ,  prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?

Exercise 11.2 | Q 65 | Page 38

If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?

Exercise 11.2 | Q 66 | Page 38

If  \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?

Exercise 11.2 | Q 67 | Page 38

If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?

Exercise 11.2 | Q 68 | Page 38

If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?

Exercise 11.2 | Q 69 | Page 38

If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?

Exercise 11.2 | Q 70 | Page 38

If \[y = \sqrt{x^2 + a^2}\] prove that  \[y\frac{dy}{dx} - x = 0\] ?

Exercise 11.2 | Q 71 | Page 38

If \[y = e^x + e^{- x}\] prove that  \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?

Exercise 11.2 | Q 72 | Page 38

If \[y = \sqrt{a^2 - x^2}\] prove that  \[y\frac{dy}{dx} + x = 0\] ?

Exercise 11.2 | Q 73 | Page 38

If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?

Exercise 11.2 | Q 74 | Page 38

Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?

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Exercise 11.3 [Pages 62 - 64]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.3 [Pages 62 - 64]

Exercise 11.3 | Q 1 | Page 62

Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?

Exercise 11.3 | Q 2 | Page 62

Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?

Exercise 11.3 | Q 3 | Page 63

Differentiate  \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\]  ?

Exercise 11.3 | Q 4 | Page 63

Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?

Exercise 11.3 | Q 5 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?

Exercise 11.3 | Q 6 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?

Exercise 11.3 | Q 7 | Page 63

Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\]  ?

Exercise 11.3 | Q 8 | Page 63

Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?

Exercise 11.3 | Q 9 | Page 63

Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?

Exercise 11.3 | Q 10 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?

Exercise 11.3 | Q 11 | Page 63

Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?

Exercise 11.3 | Q 12 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?

Exercise 11.3 | Q 13 | Page 63

Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?

Exercise 11.3 | Q 14 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?

Exercise 11.3 | Q 15 | Page 63

Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?

Exercise 11.3 | Q 16 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?

Exercise 11.3 | Q 17 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?

Exercise 11.3 | Q 18 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?

Exercise 11.3 | Q 19 | Page 63

Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?

Exercise 11.3 | Q 20 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?

Exercise 11.3 | Q 21 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?

Exercise 11.3 | Q 22 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?

Exercise 11.3 | Q 23 | Page 63

Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?

Exercise 11.3 | Q 24 | Page 63

Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?

Exercise 11.3 | Q 25 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?

Exercise 11.3 | Q 26 | Page 63

Differentiate  \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?

Exercise 11.3 | Q 27 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?

Exercise 11.3 | Q 28 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?

Exercise 11.3 | Q 29 | Page 63

 Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?

Exercise 11.3 | Q 30 | Page 63

Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?

Exercise 11.3 | Q 31 | Page 64

Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?

Exercise 11.3 | Q 32 | Page 63

Differentiate 

\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?

Exercise 11.3 | Q 33 | Page 64

Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?

Exercise 11.3 | Q 34 | Page 64

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?

Exercise 11.3 | Q 35 | Page 64

If  \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that  \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?

 

Exercise 11.3 | Q 36 | Page 64

If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that  \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?

 

Exercise 11.3 | Q 37.1 | Page 64

Differentiate the following with respect to x

\[\cos^{- 1} \left( \sin x \right)\]

Exercise 11.3 | Q 37.2 | Page 64

Differentiate the following with respect to x

\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]

Exercise 11.3 | Q 38 | Page 64

If  \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\],  show that \[\frac{dy}{dx}\] is independent of x. ? 

 

Exercise 11.3 | Q 39 | Page 64

If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ? 

Exercise 11.3 | Q 40 | Page 64

If  \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?

 

Exercise 11.3 | Q 41 | Page 64

If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?

Exercise 11.3 | Q 42 | Page 64

If  \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, 0 < x < \frac{1}{2}, \text{ find } \frac{dy}{dx} .\] ?

Exercise 11.3 | Q 43 | Page 64

If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?

Exercise 11.3 | Q 44 | Page 64

If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?

Exercise 11.3 | Q 45 | Page 64

If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?

Exercise 11.3 | Q 46 | Page 64

If \[y = \cos^{- 1} \left\{ \frac{2x - 3 \sqrt{1 - x^2}}{\sqrt{13}} \right\}, \text{ find } \frac{dy}{dx}\] ?

Exercise 11.3 | Q 47 | Page 64

Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\]  with respect to x ?

Exercise 11.3 | Q 48 | Page 64

If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?

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Exercise 11.4 [Pages 74 - 75]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.4 [Pages 74 - 75]

Exercise 11.4 | Q 1 | Page 74

Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\]  ?

Exercise 11.4 | Q 2 | Page 74

Find  \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?

 

Exercise 11.4 | Q 3 | Page 74

Find  \[\frac{dy}{dx}\] in the following case  \[x^{2/3} + y^{2/3} = a^{2/3}\] ?

 

Exercise 11.4 | Q 4 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?

 

Exercise 11.4 | Q 5 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?

Exercise 11.4 | Q 6 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?

 

Exercise 11.4 | Q 7 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?

 

Exercise 11.4 | Q 8 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?

 

Exercise 11.4 | Q 9 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?

 

Exercise 11.4 | Q 10 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?

 

Exercise 11.4 | Q 11 | Page 74

Find  \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?

 

Exercise 11.4 | Q 12 | Page 74

If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?

Exercise 11.4 | Q 13 | Page 75

If \[y \sqrt{1 - x^2} + x \sqrt{1 - y^2} = 1\] ,prove that \[\frac{dy}{dx} = - \sqrt{\frac{1 - y^2}{1 - x^2}}\] ?

Exercise 11.4 | Q 14 | Page 75

If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?

Exercise 11.4 | Q 15 | Page 75

If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?

Exercise 11.4 | Q 16 | Page 75

If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\]  ?

Exercise 11.4 | Q 17 | Page 75

If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?

Exercise 11.4 | Q 18 | Page 75

If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that  \[\frac{dy}{dx} = \frac{y}{x}\] ?

Exercise 11.4 | Q 19 | Page 75

If \[\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a\] Prove that  \[\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}\] ?

Exercise 11.4 | Q 20 | Page 75

If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?

Exercise 11.4 | Q 21 | Page 75

If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?

Exercise 11.4 | Q 22 | Page 75

If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?

Exercise 11.4 | Q 23 | Page 75

If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?

Exercise 11.4 | Q 24 | Page 75

If \[y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)\] ,Show that \[\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0\] ?

Exercise 11.4 | Q 25 | Page 75

If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find}  \frac{dy}{dx}\] ?

Exercise 11.4 | Q 26 | Page 75

If  \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find}  \frac{dy}{dx}\] ?

Exercise 11.4 | Q 27 | Page 75

If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?

Exercise 11.4 | Q 28 | Page 75

If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?

Exercise 11.4 | Q 29 | Page 75

If \[\sin^2 y + \cos xy = k,\] find  \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\] 

Exercise 11.4 | Q 30 | Page 75

If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?

Exercise 11.4 | Q 31 | Page 75

If \[\sqrt{y + x} + \sqrt{y - x} = c, \text {show that } \frac{dy}{dx} = \frac{y}{x} - \sqrt{\frac{y^2}{x^2} - 1}\] ?

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Exercise 11.5 [Pages 88 - 90]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.5 [Pages 88 - 90]

Exercise 11.5 | Q 1 | Page 88

Differentiate \[x^{1/x}\] ?

Exercise 11.5 | Q 2 | Page 88

Differentiate \[x^{\sin x}\]  ?

Exercise 11.5 | Q 3 | Page 88

Differentiate \[\left( 1 + \cos x \right)^x\] ?

Exercise 11.5 | Q 4 | Page 88

Differentiate \[x^{\cos^{- 1} x}\] ?

Exercise 11.5 | Q 5 | Page 88

Differentiate \[\left( \log x \right)^x\] ?

Exercise 11.5 | Q 6 | Page 88

Differentiate \[\left( \log x \right)^{\cos x}\] ?

Exercise 11.5 | Q 7 | Page 88

Differentiate \[\left( \sin x \right)^{\cos x}\] ?

Exercise 11.5 | Q 8 | Page 88

Differentiate \[e^{x \log x}\] ?

Exercise 11.5 | Q 9 | Page 88

Differentiate  \[\left( \sin x \right)^{\log x}\] ?

Exercise 11.5 | Q 10 | Page 88

Differentiate \[{10}^{ \log \sin x }\] ?

Exercise 11.5 | Q 11 | Page 88

Differentiate \[\left( \log x \right)^{ \log x }\] ?

Exercise 11.5 | Q 12 | Page 88

Differentiate \[{10}^\left( {10}^x \right)\] ?

Exercise 11.5 | Q 13 | Page 88

Differentiate  \[\sin \left( x^x \right)\] ?

Exercise 11.5 | Q 14 | Page 88

Differentiate \[\left( \sin^{- 1} x \right)^x\] ?

Exercise 11.5 | Q 15 | Page 88

Differentiate \[x^{\sin^{- 1} x}\]  ?

Exercise 11.5 | Q 16 | Page 88

Differentiate \[\left( \tan x \right)^{1/x}\] ?

Exercise 11.5 | Q 17 | Page 88

Differentiate \[x^{\tan^{- 1} x }\]  ?

Exercise 11.5 | Q 18.1 | Page 88

Differentiate  \[\left( x^x \right) \sqrt{x}\] ?

Exercise 11.5 | Q 18.2 | Page 88

Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?

Exercise 11.5 | Q 18.3 | Page 88

Differentiate  \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\]  ?

Exercise 11.5 | Q 18.4 | Page 88

Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?

Exercise 11.5 | Q 18.5 | Page 88

Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?

Exercise 11.5 | Q 18.6 | Page 88

Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?

Exercise 11.5 | Q 18.7 | Page 88

Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?

Exercise 11.5 | Q 18.8 | Page 88

Differentiate  \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?

Exercise 11.5 | Q 19 | Page 89

Find  \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?

 

Exercise 11.5 | Q 20 | Page 89
Find \[\frac{dy}{dx}\]  \[y = x^n + n^x + x^x + n^n\] ?
Exercise 11.5 | Q 21 | Page 89

find  \[\frac{dy}{dx}\]  \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?

 

Exercise 11.5 | Q 22 | Page 89

Find  \[\frac{dy}{dx}\]  \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?

 

Exercise 11.5 | Q 23 | Page 89

Find  \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?

 

Exercise 11.5 | Q 24 | Page 89

Find  \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?

 

Exercise 11.5 | Q 25 | Page 89

Find  \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?

Exercise 11.5 | Q 26 | Page 89

Find  \[\frac{dy}{dx}\]  \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?

 

Exercise 11.5 | Q 27 | Page 89

Find \[\frac{dy}{dx}\] \[y =  \left( \tan  x \right)^{\cot   x}  +  \left( \cot  x \right)^{\tan  x}\] ?

Exercise 11.5 | Q 28 | Page 89

Fine \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^x + \sin^{- 1} \sqrt{x}\] ?

Exercise 11.5 | Q 29.1 | Page 89

Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?

Exercise 11.5 | Q 29.2 | Page 89

Find \[\frac{dy}{dx}\]  \[y = x^x + \left( \sin x \right)^x\] ?

Exercise 11.5 | Q 30 | Page 89

Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?

Exercise 11.5 | Q 31 | Page 89

Find \[\frac{dy}{dx}\]

\[y = x^x + x^{1/x}\] ?

Exercise 11.5 | Q 32 | Page 89

Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?

Exercise 11.5 | Q 33 | Page 89

If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?

Exercise 11.5 | Q 34 | Page 89

If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?

Exercise 11.5 | Q 35 | Page 89

If \[y = \sin \left( x^x \right)\] prove that  \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?

Exercise 11.5 | Q 36 | Page 89

If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?

Exercise 11.5 | Q 37 | Page 89

If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?

Exercise 11.5 | Q 38 | Page 89

If \[x^y + y^x = \left( x + y \right)^{x + y} , \text{ find } \frac{dy}{dx}\] ?

Exercise 11.5 | Q 39 | Page 89

If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?

Exercise 11.5 | Q 40 | Page 89

If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?

Exercise 11.5 | Q 41 | Page 89

If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?

Exercise 11.5 | Q 42 | Page 89

If \[\left( \cos x \right)^y = \left( \tan y \right)^x\] , prove that \[\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }\] ?

Exercise 11.5 | Q 43 | Page 89

If \[e^x + e^y = e^{x + y}\] , prove that

\[\frac{dy}{dx} + e^{y - x} = 0\] ?

Exercise 11.5 | Q 44 | Page 90

If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?

Exercise 11.5 | Q 45 | Page 90

If \[e^{x + y} - x = 0\] ,prove that \[\frac{dy}{dx} = \frac{1 - x}{x}\] ?

Exercise 11.5 | Q 46 | Page 90

If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?

 

Exercise 11.5 | Q 47 | Page 90

If  \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?

 

Exercise 11.5 | Q 48 | Page 90

If  \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?

 

Exercise 11.5 | Q 49 | Page 90

If \[xy \log \left( x + y \right) = 1\] , prove that  \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?

Exercise 11.5 | Q 50 | Page 90

If \[y = x \sin y\] , prove that  \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?

 

Exercise 11.5 | Q 51 | Page 90

Find the derivative of the function f (x) given by  \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?

 

Exercise 11.5 | Q 52 | Page 90

If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?

Exercise 11.5 | Q 53 | Page 90

If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?

Exercise 11.5 | Q 54 | Page 90

If  \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?

 

Exercise 11.5 | Q 55 | Page 90

If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?

Exercise 11.5 | Q 56 | Page 90

If  \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?

 

Exercise 11.5 | Q 57 | Page 90
\[\text{ If }\cos y = x\cos\left( a + y \right),\text{  where } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Exercise 11.5 | Q 58 | Page 90
\[\text{ If } \left( x - y \right) e^\frac{x}{x - y} = a,\text{  prove that y }\frac{dy}{dx} + x = 2y\] ?
Exercise 11.5 | Q 59 | Page 90
\[\text{ If } x = e^{x/y} , \text{ prove that } \frac{dy}{dx} = \frac{x - y}{x\log x}\] ?
Exercise 11.5 | Q 60 | Page 90
\[\text{ If }y = x^{\tan x} + \sqrt{\frac{x^2 + 1}{2}}, \text{ find} \frac{dy}{dx}\] ?

 

Exercise 11.5 | Q 61 | Page 90
\[\text{ If y } = 1 + \frac{\alpha}{\left( \frac{1}{x} - \alpha \right)} + \frac{{\beta}/{x}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)} + \frac{{\gamma}/{x^2}}{\left( \frac{1}{x} - \alpha \right)\left( \frac{1}{x} - \beta \right)\left( \frac{1}{x} - \gamma \right)}, \text{ find } \frac{dy}{dx} \] ?
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Exercise 11.6 [Pages 98 - 99]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.6 [Pages 98 - 99]

Exercise 11.6 | Q 1 | Page 98

If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?

Exercise 11.6 | Q 2 | Page 98

If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?

Exercise 11.6 | Q 3 | Page 98

If  \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?

 

Exercise 11.6 | Q 4 | Page 98

If  \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?

 

Exercise 11.6 | Q 5 | Page 98

\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?

Exercise 11.6 | Q 6 | Page 98

If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?

 

Exercise 11.6 | Q 7 | Page 99

If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that  \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]

 

Exercise 11.6 | Q 8 | Page 99

If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?

 

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Exercise 11.7 [Pages 103 - 104]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.7 [Pages 103 - 104]

Exercise 11.7 | Q 1 | Page 103

Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?

Exercise 11.7 | Q 2 | Page 103

Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?

Exercise 11.7 | Q 3 | Page 103

If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?

Exercise 11.7 | Q 4 | Page 103

Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?

Exercise 11.7 | Q 5 | Page 103

Find \[\frac{dy}{dx}\] , when \[x = b   \sin^2   \theta  \text{ and }  y = a   \cos^2   \theta\] ?

Exercise 11.7 | Q 6 | Page 103

Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta  = \frac{\pi}{2}\] ?

Exercise 11.7 | Q 7 | Page 103

Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?

Exercise 11.7 | Q 8 | Page 103

Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?

Exercise 11.7 | Q 9 | Page 103

Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?

Exercise 11.7 | Q 10 | Page 103

Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?

Exercise 11.7 | Q 11 | Page 103

Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?

Exercise 11.7 | Q 12 | Page 103

Find \[\frac{dy}{dx}\] , when  \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?

Exercise 11.7 | Q 13 | Page 103

Find  \[\frac{dy}{dx}\] , when  \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?

 

Exercise 11.7 | Q 14 | Page 103

If  \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?

Exercise 11.7 | Q 15 | Page 103

If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?

Exercise 11.7 | Q 16 | Page 103

If \[x = \cos t \text{ and y }  = \sin t,\] prove that  \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?

 

Exercise 11.7 | Q 17 | Page 103

If  \[x = a\left( t + \frac{1}{t} \right) \text{ and y } = a\left( t - \frac{1}{t} \right)\] ,prove that  \[\frac{dy}{dx} = \frac{x}{y}\]?

 

Exercise 11.7 | Q 18 | Page 103
If \[x = \sin^{- 1} \left( \frac{2 t}{1 + t^2} \right) \text{ and y } = \tan^{- 1} \left( \frac{2 t}{1 - t^2} \right), - 1 < t < 1\] porve that \[\frac{dy}{dx} = 1\] ?

 

Exercise 11.7 | Q 19 | Page 103

If  \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?

 

Exercise 11.7 | Q 20 | Page 103

If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?

Exercise 11.7 | Q 21 | Page 103

If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?

Exercise 11.7 | Q 22 | Page 104

If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?

 

Exercise 11.7 | Q 23 | Page 104

If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?

 

Exercise 11.7 | Q 24 | Page 104

If  \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at  \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?

Exercise 11.7 | Q 25 | Page 104

\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?

Exercise 11.7 | Q 26 | Page 104

If  \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?

Exercise 11.7 | Q 27 | Page 104
\[\sin x = \frac{2t}{1 + t^2}, \tan y = \frac{2t}{1 - t^2}, \text { find }  \frac{dy}{dx}\] ?
Exercise 11.7 | Q 28 | Page 104

Write the derivative of sinx with respect to cos x ?

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Exercise 11.8 [Pages 112 - 113]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation Exercise 11.8 [Pages 112 - 113]

Exercise 11.8 | Q 1 | Page 112

Differentiate x2 with respect to x3

Exercise 11.8 | Q 2 | Page 112

Differentiate log (1 + x2) with respect to tan−1 x ?

Exercise 11.8 | Q 3 | Page 112

Differentiate (log x)x with respect to log x ?

Exercise 11.8 | Q 4.1 | Page 112

Differentiate  \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\]  ?

 

Exercise 11.8 | Q 4.2 | Page 112

Differentiate  \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?

Exercise 11.8 | Q 5.1 | Page 112
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2 \sqrt{2}}, \frac{1}{\sqrt{2 \sqrt{2}}} \right)\] ?
Exercise 11.8 | Q 5.2 | Page 112

Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?

Exercise 11.8 | Q 5.3 | Page 112

Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?

Exercise 11.8 | Q 6 | Page 112

Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?

Exercise 11.8 | Q 7.1 | Page 112

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to  \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?

Exercise 11.8 | Q 7.2 | Page 112

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to  \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?

Exercise 11.8 | Q 8 | Page 112

Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?

Exercise 11.8 | Q 9 | Page 112

Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1\] ?

Exercise 11.8 | Q 10 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?

Exercise 11.8 | Q 11 | Page 113

Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?

Exercise 11.8 | Q 12 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text {  if }0 < x < 1\] ?

Exercise 11.8 | Q 13 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)\] with respect to \[\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}\] ?

Exercise 11.8 | Q 14 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with  respect to \[\sec^{- 1} x\] ?

Exercise 11.8 | Q 15 | Page 113

Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?

Exercise 11.8 | Q 16 | Page 113

Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ? 

Exercise 11.8 | Q 17 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?

Exercise 11.8 | Q 18 | Page 113

\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ? 

Exercise 11.8 | Q 19 | Page 113

Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?

Exercise 11.8 | Q 20 | Page 113

Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?

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[Pages 116 - 118]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation [Pages 116 - 118]

Q 1 | Page 117

If f (x) = loge (loge x), then write the value of `f' (e)` ?

Q 2 | Page 117

If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?

Q 3 | Page 117

If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?

Q 4 | Page 117

If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of  \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?

 

Q 5 | Page 117

If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?

Q 6 | Page 117

Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.

Q 7 | Page 117

If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?

Q 8 | Page 117

If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?

Q 9 | Page 117

If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?

Q 10 | Page 118

If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?

Q 11 | Page 118

If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?

Q 12 | Page 118

If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?

Q 13 | Page 118

If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?

Q 14 | Page 118

If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?

Q 15 | Page 118

If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?

Q 16 | Page 116

If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?

Q 17 | Page 118

If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\]  ?

Q 18 | Page 118

If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ? 

Q 19 | Page 118

If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?

Q 20 | Page 118

If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?

Q 21 | Page 118

If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?

Q 22 | Page 118

If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \]  to ∞, then find the value of  \[\frac{dy}{dx}\] ?

Q 23 | Page 118

If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?

Q 24 | Page 118

If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?

Q 25 | Page 118

If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ? 

Q 26 | Page 118

If f (x) is an even function, then write whether `f' (x)` is even or odd ?

Q 27 | Page 118

If f (x) is an odd function, then write whether `f' (x)` is even or odd ?

Q 28 | Page 118

If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?

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[Pages 119 - 122]

RD Sharma solutions for Class 12 Maths Chapter 11 Differentiation [Pages 119 - 122]

Q 1 | Page 119

If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .

  • 0

  • 1

  • 1/e

  • 1/2e

Q 2 | Page 119

The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .

  • \[\frac{x}{\log x}\]

  • \[\frac{\log x}{x}\]

  • \[\left( x \log x \right)^{- 1}\]

  • none of these

Q 3 | Page 119

The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .

  • (2/3)1/2

  • (1/3)1/2

  • 31/2

  • 61/2

Q 4 | Page 119

Differential coefficient of sec \[\sec \left( \tan^{- 1} x \right)\] is _____________ .

  • \[\frac{x}{1 + x^2}\]

  • \[x \sqrt{1 + x^2}\]

  • \[\frac{1}{\sqrt{1 + x^2}}\]

  • \[\frac{x}{\sqrt{1 + x^2}}\]

Q 5 | Page 119

If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\] _________ .

  • − 1/4

  • − 1/2

  • 1/4

  • 1/2

Q 6 | Page 119

If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{ then} \frac{dy}{dx} =\] ____________ .

  • \[\left( 1 + \frac{1}{x} \right)^x \left( 1 + \frac{1}{x} \right) - \frac{1}{x + 1}\]

  • \[\left( 1 + \frac{1}{x} \right)^x \log \left( 1 + \frac{1}{x} \right)\]

  • \[\left( x + \frac{1}{x} \right)^x \left\{ \log \left( x + 1 \right) - \frac{x}{x + 1} \right\}\]

  • \[\left( x + \frac{1}{x} \right)^x \left\{ \log \left( 1 + \frac{1}{x} \right) + \frac{1}{x + 1} \right\}\]

Q 7 | Page 119

If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .

  • \[\frac{1 + x}{1 + \log x}\]

  • \[\frac{1 - \log x}{1 + \log x}\]

  • not defined

  • \[\frac{\log x}{\left( 1 + \log x \right)^2}\]

Q 8 | Page 119

Given  \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .

  • \[f'\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)\]

  • \[f\left( \frac{1}{2} \right) = - f'\left( - \frac{1}{2} \right)\]

  • \[f\left( - \frac{1}{2} \right) = f\left( - \frac{1}{2} \right)\]

  • \[f\left( \frac{1}{2} \right) = f'\left( - \frac{1}{2} \right)\]

Q 9 | Page 119

If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .

  • \[\tan^2 \theta\]

  • \[\sec^2 \theta\]

  • \[\sec \theta\]

  • \[\left| \sec \theta \right|\]

Q 10 | Page 120

If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .

  • \[- \frac{2}{1 + x^2}\]

  • \[\frac{2}{1 + x^2}\]

  • \[\frac{1}{2 - x^2}\]

  • \[\frac{2}{2 - x^2}\]

Q 11 | Page 120

The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]

  • does not exist

  • 0

  • 1/2

  • 1/3

Q 12 | Page 120

For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text {  at } \left( 1/4, 1/4 \right)\text {  is }\] _____________ .

  • 1/2

  • 1

  • -1

  • 2

Q 13 | Page 120

If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .

  • 2

  • -2

  • 1

  • -1

Q 14 | Page 120

Let  \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .

  • 1/2

  • x

  • \[\frac{1 - x^2}{x^2 - 4}\]

  • 1

Q 15 | Page 120

\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .

  • 1/2

  • -1/2

  • 1

  • -1

Q 16 | Page 120
\[\frac{d}{dx} \left[ \log \left\{ e^x \left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right]\] equals ___________ .
  • \[\frac{x^2 - 1}{x^2 - 4}\]

  • 1

  • \[\frac{x^2 + 1}{x^2 - 4}\]

  • \[e^x \frac{x^2 - 1}{x^2 - 4}\]

Q 17 | Page 120

If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .

  • \[\frac{\sin x}{2 y - 1}\]

  • \[\frac{\sin x}{1 - 2 y}\]

  • \[\frac{\cos x}{1 - 2 y}\]

  • \[\frac{\cos x}{2 y - 1}\]

Q 18 | Page 120

If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .

  • \[- \frac{y}{x}\]

  • \[\frac{3 \sin \left( xy \right) + 4 \cos \left( xy \right)}{3 \cos \left( xy \right) - 4 \sin \left( xy \right)}\]

  • \[\frac{3 \cos \left( xy \right) + 4 \sin \left( xy \right)}{4 \cos \left( xy \right) - 3 \sin \left( xy \right)}\]

  • none of these

Q 19 | Page 120

If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .

  • \[\frac{\sin a}{\sin a \sin^2 \left( a + y \right)}\]

  • \[\frac{\sin^2 \left( a + y \right)}{\sin a}\]

  • \[\sin a \sin^2 \left( a + y \right)\]

  • \[\frac{\sin^2 \left( a - y \right)}{\sin a}\]

Q 20 | Page 121

The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to  \[\cos^{- 1} x\]  is ___________ .

  • `2`

  • \[\frac{1}{2 \sqrt{1 - x^2}}\]

  • \[2/x\]

  • \[1 - x^2\]

Q 21 | Page 121

If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .

  • \[1 \text { for x } < - 3\]

  • \[- 1\text {  for x} < - 3\]

  • \[1\text {  for all } x \in R\]

  • none of these

Q 22 | Page 121

If \[f\left( x \right) = \left| x^2 - 9x + 20 \right|\]  then `f' (x)` is equal to ____________ .

  • \[- 2x + 9\text {  for all } x \in R\]

  • \[2x - 9 \text { if }4 < x < 5\]

  • \[- 2x + 9, \text { if }4 < x < 5\]

  • none of these

Q 23 | Page 121

If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\]  then the derivative of f (x) in the interval [0, 7] is ____________ .

  • 1

  • -1

  • 0

  • none of these

Q 24 | Page 121

If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\]  is equal to __________ .

  • 1

  • -1

  • 0

  • none of these

Q 25 | Page 121

If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to _____________ .

  • 1

  • 0

  • \[x^{l + m + n}\]

  • none of these

Q 26 | Page 121

If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\]  is equal to ______________ .

  • 1

  • \[\left( a + b + c \right)^{x^{a + b + c - 1}}\]

  • 0

  • none of these

Q 27 | Page 121

If  \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .

  • \[\frac{x^2}{y^2} \sqrt{\frac{1 - y^6}{1 - x^6}}\]

  • \[\frac{y^2}{x^2}\sqrt{\frac{1 - y^6}{1 + x^6}}\]

  • \[\frac{x^2}{y^2}\sqrt{\frac{1 - x^6}{1 - y^6}}\]

  • none of these

Q 28 | Page 121

If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .

  • 1

  • 0

  • `1/2`

Q 29 | Page 121

If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .

  • \[\frac{x^2 - y^2}{x^2 + y^2}\]

  • `y/x`

  • `x/y`

  • none of these

Q 30 | Page 121

If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .

  • \[\frac{\cos^2 \left( a + y \right)}{\cos a}\]

  • \[\frac{\cos a}{\cos^2 \left( a + y \right)}\]

  • \[\frac{\sin^2 y}{\cos a}\]

  • none of these

Q 31 | Page 122

If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .

  • \[\frac{4 x^3}{1 - x^4}\]

  • \[- \frac{4x}{1 - x^4}\]

  • \[\frac{1}{4 - x^4}\]

  • \[- \frac{4 x^3}{1 - x^4}\]

Q 32 | Page 122

If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .

  • \[\frac{\cos x}{2y - 1}\]

  • \[\frac{\cos x}{1 - 2y}\]

  • \[\frac{\sin x}{1 - 2y}\]

  • \[\frac{\sin x}{2y - 1}\]

Q 33 | Page 122

If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then  } \frac{dy}{dx}\] is equal to ___________ .

  • `1/2`

  • 0

  • 1

  • none of these

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Chapter 11: Differentiation

Exercise 11.1Exercise 11.2Exercise 11.3Exercise 11.4Exercise 11.5Exercise 11.6Exercise 11.7Exercise 11.8Others
Class 12 Maths - Shaalaa.com

RD Sharma solutions for Class 12 Maths chapter 11 - Differentiation

RD Sharma solutions for Class 12 Maths chapter 11 (Differentiation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12 Maths chapter 11 Differentiation are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

Using RD Sharma Class 12 solutions Differentiation exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 11 Differentiation Class 12 extra questions for Class 12 Maths and can use Shaalaa.com to keep it handy for your exam preparation

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