Advertisements
Advertisements
प्रश्न
Differentiate (log x)x with respect to log x ?
उत्तर
\[ \Rightarrow \frac{1}{u}\frac{du}{dx} = x\left( \frac{1}{\log x} \right)\frac{d}{dx}\left( \log x \right) + \log\log x\left( 1 \right)\]
\[ \Rightarrow \frac{du}{dx} = u\left[ \frac{x}{\log x}\left( \frac{1}{x} \right) + \log \log x \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( \log x \right)^x \left[ \frac{1}{\log x} + \log \log x \right] . . \left( i \right)\]
\[\text { Again, let v } = \log x\]
\[ \Rightarrow \frac{dv}{dx} = \frac{1}{x} . . . \left( ii \right)\]
\[\text { Dividing equation } \left( i \right) \text { by } \left( ii \right), \text { we get }\]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{\left( \log x \right)^x \left[ \frac{1}{\log x} + \log \log x \right]}{\frac{1}{x}}\]
\[ \Rightarrow \frac{du}{dv} = \frac{\left( \log x \right)^x \left[ \frac{1 + \log x\left( \log \log x \right)}{\log x} \right]}{\frac{1}{x}}\]
\[ \Rightarrow \frac{du}{dv} = x \left( \log x \right)^{x^{- 1}} \left( 1 + \log x \times \log \log x \right)\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate sin (3x + 5) ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{3 x} \cos 2x\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
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 \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\] with respect to x ?
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 \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
Differential coefficient of sec(tan−1 x) is ______.
If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
Find the second order derivatives of the following function sin (log x) ?
If y = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
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 = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 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 \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If y = a + bx2, a, b arbitrary constants, then
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
