Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
Advertisements
उत्तर
\[\text{ We have, y } = x^x + x^\frac{1}{x} \]
\[ \Rightarrow y = e^{\log x^x }+ e^{\log x^{\frac{1}{x}} }\]
\[ \Rightarrow y = e^{x \log x} + e^\left( \frac{1}{x}\log x \right)\]
Differentiating with respect to x using chain rule,
\[\frac{dy}{dx} = \frac{d}{dx}\left( e^{x \log x} \right) + \frac{d}{dx}\left( e^{\frac{1}{x}\log x }\right)\]
\[ = e^{x \log x} \frac{d}{dx}\left( x \log x \right) + e^{\frac{1}{x}\log x} \frac{d}{dx}\left( \frac{1}{x}\log x \right)\]
\[ = e^{\log x^x} \left[ x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right) \right] + e^{ \log x^{\frac{1}{x}}} \left[ \frac{1}{x}\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( \frac{1}{x} \right) \right] \]
\[ = \left( x \right)^x \left[ x\left( \frac{1}{x} \right) + \log x\left( 1 \right) \right] + x^\frac{1}{x} \left[ \left( \frac{1}{x} \right)\left( \frac{1}{x} \right) + \log x\left( - \frac{1}{x^2} \right) \right]\]
\[ = \left( x \right)^x \left[ 1 + \log x \right] + x^\frac{1}{x} \left( \frac{1}{x^2} - \frac{1}{x^2}\log x \right)\]
\[ = x^x \left[ 1 + \log x \right] + x^\frac{1}{x} \left( \frac{1 - \log x}{x^2} \right)\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cos x ?
Differentiate tan (x° + 45°) ?
Differentiate \[e^{3 x} \cos 2x\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
Differentiate \[x^{\sin x}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
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 x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
