Advertisements
Advertisements
Question
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
Advertisements
Solution
\[\text{Let, y} = \tan^{- 1} \left( \frac{x - a}{x + a} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{\frac{x - a}{x}}{\frac{x + a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{\frac{x}{x} - \frac{a}{x}}{\frac{x}{x} + \frac{a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{1 - \frac{a}{x}}{1 + 1 \times \frac{a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( 1 \right) - \tan^{- 1} \left( \frac{a}{x} \right)\]
Differentiate it with respect to x,
\[\frac{d y}{d x} = 0 - \frac{1}{1 + \left( \frac{a}{x} \right)^2}\frac{d}{dx}\left( \frac{a}{x} \right)\]
\[ \Rightarrow \frac{d y}{d x} = - \frac{x^2}{x^2 + a^2}\left( \frac{- a}{x^2} \right)\]
\[ \therefore \frac{d y}{d x} = \frac{a}{a^2 + x^2}\]
APPEARS IN
RELATED QUESTIONS
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
Differentiate tan 5x° ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
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 \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
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 \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
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 \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function x3 log x ?
Find the second order derivatives of the following function tan−1 x ?
Find the second order derivatives of the following function x cos x ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
f(x) = 3x2 + 6x + 8, x ∈ R
