Advertisements
Advertisements
Question
Options
\[\frac{x^2 - 1}{x^2 - 4}\]
1
\[\frac{x^2 + 1}{x^2 - 4}\]
\[e^x \frac{x^2 - 1}{x^2 - 4}\]
Advertisements
Solution
\[\frac{x^2 - 1}{x^2 - 4}\]
\[\text { Let y } = \frac{d}{dx}\left[ \log\left\{ e^x \left( \frac{x - 2}{x + 2} \right)^\frac{3}{4} \right\} \right]\]
\[ \Rightarrow y = \frac{d}{dx}\left[ x\log e + \frac{3}{4}\log\left( \frac{x - 2}{x + 2} \right) \right]\]
\[ \Rightarrow y = \frac{d}{dx}\left[ x + \frac{3}{4}\log\left( \frac{x - 2}{x + 2} \right) \right]\]
\[ \Rightarrow \frac{dy}{dx} = 1 + \frac{3}{4\left( \frac{x - 2}{x + 2} \right)} \times \frac{\left( x + 2 \right) \times 1 - \left( x - 2 \right) \times 1}{\left( x + 2 \right)^2}\]
\[ \Rightarrow \frac{dy}{dx} = 1 + \frac{3\left( x + 2 \right)}{4\left( x - 2 \right)} \times \frac{x + 2 - x + 2}{\left( x + 2 \right)^2}\]
\[ \Rightarrow \frac{dy}{dx} = 1 + \frac{3\left( x + 2 \right)}{4\left( x - 2 \right)} \times \frac{4}{\left( x + 2 \right)}\]
\[ \Rightarrow \frac{dy}{dx} = 1 + \frac{3}{\left( x^2 - 4 \right)}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x^2 - 4 + 3}{x^2 - 4}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{x^2 - 1}{x^2 - 4}\]
APPEARS IN
RELATED QUESTIONS
Differentiate tan2 x ?
Differentiate tan (x° + 45°) ?
Differentiate (log sin x)2 ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
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 \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
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 \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?
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 \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
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 (x) is an even function, then write whether `f' (x)` is even or odd ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
Find the second order derivatives of the following function tan−1 x ?
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
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 = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
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.
