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

Differentiate the following functions from first principles e^−x.

Differentiate the following functions from first principles e^3x.

Differentiate the following functions from first principles e^ax+b.

Differentiate the following functions from first principles e^cos x.

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

Differentiate the following functions from first principles log cos x ?

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

Differentiate the following functions from first principles x²e^x ?

Differentiate the following functions from first principles log cosec x ?

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

Differentiate sin (3x + 5) ?

Differentiate tan² x ?

Differentiate tan (x° + 45°) ?

Differentiate sin (log x) ?

Differentiate \[e^{\sin} \sqrt{x}\] ?

Differentiate e^tan x ?

Differentiate sin² (2x + 1) ?

Differentiate log₇ (2x − 3) ?

Differentiate tan 5x° ?

Differentiate `2^(x^3)` ?

Differentiate \[3^{e^x}\] ?

Differentiate log_x 3 ?

Differentiate \[3^{x^2 + 2x}\] ?

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

Differentiate \[3^{x \log x}\] ?

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

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

Differentiate (log sin x)² ?

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

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

Differentiate \[e^{3 x} \cos 2x\] ?

Differentiate sin(log sin x) ?

Differentiate \[e^{\tan 3 x} \] ?

Differentiate \[e^\sqrt{\cot x}\] ?

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

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

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

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

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

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

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

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

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

Differentiate \[e^{\sin^{- 1} 2x}\] ?

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

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

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

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

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

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

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

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

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

Differentiate  \[e^x \log \sin 2x\] ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Differentiate 

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

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

Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] with respect to x.

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  \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?

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  \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?

Differentiate the following with respect to x: 

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

Differentiate the following with respect to x: 

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

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. ? 

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 \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ? 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`

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

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

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

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

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

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

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

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

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

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

If  y `sqrt(x^2 + 1) = log sqrt(x^2 + 1) - x`, show that `(x^2 + 1)(dy)/(dx) + xy + 1 = 0.`

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

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

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

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

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

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

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

Differentiate \[x^{1/x}\]  with respect to x.

Differentiate \[x^{\sin x}\]  ?

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

Differentiate \[x^{\cos^{- 1} x}\] ?

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

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

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

Differentiate \[e^{x \log x}\] ?

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

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

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

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

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

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

Differentiate \[x^{\sin^{- 1} x}\]  ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

If `y=(sinx)^x + sin^-1 sqrtx  "then find"  dy/dx` 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)` ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Differentiate x² with respect to x³

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

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

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

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

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

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

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 \[- 1 < x < 1, x \neq 0 .\] ?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ?

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

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)`.

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

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

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

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

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 \[\frac{dy}{dx} \text{ at x } = 0\] ?

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

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

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

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

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

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

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

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

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

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

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

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 \[\frac{du}{dv}\] ?

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)` ?

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

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

If f (x) = log_x² (log x), the `f' (x)` at x = e is ____________ .

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

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

Differential coefficient of sec(tan⁻¹ x) is ______.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 _____________ .

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 ______________ .

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

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

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

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

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

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

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