Advertisements
Advertisements
प्रश्न
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 \[- 1 < x < 1, x \neq 0 .\] ?
उत्तर
\[\text { Let, u }= \tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\]
\[\text { put x }= \tan\theta\]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{\sqrt{1 + \tan^2 \theta} - 1}{\tan\theta} \right)\]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{sec\theta - 1}{\tan\theta} \right) \]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{1 - \cos\theta}{\sin\theta} \right) \]
\[ \Rightarrow u = \tan^{- 1} \left( \frac{2 \sin^2 \frac{\theta}{2}}{2\sin\frac{\theta}{2}\cos\frac{\theta}{2}} \right) \]
\[ \Rightarrow u = \tan^{- 1} \left( \tan\frac{\theta}{2} \right) . . . \left( i \right)\]
\[\text { And,} \]
\[ v = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\]
\[ \Rightarrow v = \sin^{- 1} \left( \frac{2\tan\theta}{1 + \tan^2 \theta} \right) \]
\[ \Rightarrow v = \sin^{- 1} \left( \sin2\theta \right) . . . \left( ii \right)\]
\[\text { Here }, \]
\[ - 1 < x < 1\]
\[ \Rightarrow - 1 < \tan\theta < 1 \]
\[ \Rightarrow - \frac{\pi}{4} < \theta < \frac{\pi}{4} . . . \left( A \right) \]
\[\text { So, from equation } \left( i \right), \]
\[u = \frac{\theta}{2} .........\left[ \text { Since }, \tan^{- 1} \left( \tan\theta \right) = \theta, \text{ if }\theta \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \right] \]
\[ \Rightarrow u = \frac{1}{2} \tan^{- 1} x ..........\left[ \text { since, } x = \tan\theta \right]\]
Differentiating it with respect to x,
\[\frac{du}{dx} = \frac{1}{2}\left( \frac{1}{1 + x^2} \right)\]
\[ \Rightarrow \frac{du}{dx} = \frac{1}{2\left( 1 + x^2 \right)} . . . \left( i \right)\]
\[\text { Now, from equation } \left( ii \right) \text { and } \left( A \right), \]
\[v = 2\theta .........\left[ \text { Since }, \sin^{- 1} \left( \sin\theta \right) = \theta, \text{ if }\theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ \Rightarrow v = 2 \tan^{- 1} x .........\left[ \text { Since, } x = \tan\theta \right]\]
Differentiating it with respect to x,
\[\frac{dv}{dx} = 2\left( \frac{1}{1 + x^2} \right) . . . \left( iv \right)\]
\[\text { dividing equation } \left( iii \right) \text { by } \left( iv \right), \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{1}{2\left( 1 + x^2 \right)} \times \frac{1 + x^2}{2}\]
\[ \therefore \frac{du}{dv} = \frac{1}{4}\]
संबंधित प्रश्न
Differentiate \[3^{e^x}\] ?
Differentiate logx 3 ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
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 \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
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 \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
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 \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
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 \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
Differentiate x2 with respect to x3
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
Find the second order derivatives of the following function ex sin 5x ?
If x = a (1 − cos3 θ), y = a sin3 θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\] ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
