Advertisements
Advertisements
Question
Differentiate (log x)x with respect to log x ?
Advertisements
Solution
\[ \Rightarrow \frac{1}{u}\frac{du}{dx} = x\left( \frac{1}{\log x} \right)\frac{d}{dx}\left( \log x \right) + \log\log x\left( 1 \right)\]
\[ \Rightarrow \frac{du}{dx} = u\left[ \frac{x}{\log x}\left( \frac{1}{x} \right) + \log \log x \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( \log x \right)^x \left[ \frac{1}{\log x} + \log \log x \right] . . \left( i \right)\]
\[\text { Again, let v } = \log x\]
\[ \Rightarrow \frac{dv}{dx} = \frac{1}{x} . . . \left( ii \right)\]
\[\text { Dividing equation } \left( i \right) \text { by } \left( ii \right), \text { we get }\]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = \frac{\left( \log x \right)^x \left[ \frac{1}{\log x} + \log \log x \right]}{\frac{1}{x}}\]
\[ \Rightarrow \frac{du}{dv} = \frac{\left( \log x \right)^x \left[ \frac{1 + \log x\left( \log \log x \right)}{\log x} \right]}{\frac{1}{x}}\]
\[ \Rightarrow \frac{du}{dv} = x \left( \log x \right)^{x^{- 1}} \left( 1 + \log x \times \log \log x \right)\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles eax+b.
Differentiate the following functions from first principles ecos x.
Differentiate sin (3x + 5) ?
Differentiate tan 5x° ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
Differentiate \[\cos \left( \log x \right)^2\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
If \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
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 x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
f(x) = xx has a stationary point at ______.
