Advertisements
Advertisements
प्रश्न
If f (x) = loge (loge x), then write the value of `f' (e)` ?
Advertisements
उत्तर
\[\text { We have }, f\left( x \right) = \log_e \left( \log_e x \right)\]
Differentiating with respect to x,
\[f'\left( x \right) = \frac{1}{\log_e x}\frac{d}{dx}\left( \log_e x \right) \]
\[ \Rightarrow f'\left( x \right) = \frac{1}{\log_e x}\left( \frac{1}{x} \right)\]
\[ \Rightarrow f'\left( e \right) = \frac{1}{\log_e e}\left( \frac{1}{e} \right) \left[ \because x = e \right]\]
\[ \Rightarrow f'\left( e \right) = \frac{1}{e} \left[ \because \log_e e = 1 \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e3x.
Differentiate the following functions from first principles eax+b.
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate etan x ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
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 \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
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 `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
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 \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate log (1 + x2) with respect to tan−1 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 \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
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 = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
Differentiate sin(log sin x) ?
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
