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 \[- 1 < x < 1, x \neq 0 .\] ?
Advertisements
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 \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
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 \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
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 \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Find \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
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 \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
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 \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
Find the second order derivatives of the following function x3 + tan x ?
If x = a (1 − cos3θ), y = a sin3θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\]?
If x = cos θ, y = sin3 θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
