Advertisements
Advertisements
प्रश्न
Differentiate \[\left( \log x \right)^x\] ?
Advertisements
उत्तर
\[\text{Let y }= \left( \log x \right)^x . . . \left( i \right)\]
Taking log on both sides,
\[\log y = \log \left( \log x \right)^x \]
\[ \Rightarrow \log y = x\log\left( \log x \right)\]
Differentiating with respect to x using chain rule,
\[\frac{1}{y}\frac{dy}{dx} = x\frac{d}{dx}\log\left( \log x \right) + \log\left( \log x \right)\frac{d}{dx}\left( x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = x\frac{1}{\log x}\frac{d}{dx}\left( \log x \right) + \log\left( \log x \right)\left( 1 \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{x}{\log x}\left( \frac{1}{x} \right) + \log\left( \log x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = \frac{1}{\log x} + \log\left( \log x \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ \frac{1}{\log x} + \log\left( \log x \right) \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left( \log x \right)^x \left[ \frac{1}{\log x} + \log\left( \log x \right) \right] \left[ \text{using equation }\left( i \right) \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (log x) ?
Differentiate log7 (2x − 3) ?
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\]?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
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 \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
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 \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
Differential coefficient of sec(tan−1 x) is ______.
If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\] is equal to ______________ .
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
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 x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]
Disclaimer: There is a misprint in the question,
\[\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0\] must be written instead of
\[\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \] ?
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
Find the minimum value of (ax + by), where xy = c2.
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
