Advertisements
Advertisements
Question
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Advertisements
Solution
\[\text{ Let y } = x^x \sqrt{x} . . . \left( i \right)\]
\[\text{ Taking log on both sides }, \]
\[\log y = \log\left( x^x \sqrt{x} \right)\]
\[ \Rightarrow \log y = \log x^x + \log x^\frac{1}{2} \]
\[ \Rightarrow \log y = x \log x + \frac{1}{2}\log x \]
Differentiating with respect to x,
\[\frac{1}{y}\frac{dy}{dx} = x\frac{d}{dx}\left( \log x \right) + \log x\frac{d}{dx}\left( x \right) + \frac{1}{2}\frac{d}{dx}\left( \log x \right) \]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = x\left( \frac{1}{x} \right) + \log x\left( 1 \right) + \frac{1}{2}\left( \frac{1}{x} \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = 1 + \log x + \frac{1}{2x}\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ 1 + \log x + \frac{1}{2x} \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^x \sqrt{x}\left[ 1 + \log x + \frac{1}{2x} \right] \left[ \text{ using equation} \left( i \right) \right]\]
\[ \Rightarrow \frac{dy}{dx} = x^{x + \frac{1}{2}} \left[ \left( \frac{2x + 1}{2x} \right) + \log x \right]\]
APPEARS IN
RELATED QUESTIONS
Differentiate sin (log x) ?
Differentiate sin2 (2x + 1) ?
Differentiate tan 5x° ?
Differentiate logx 3 ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If \[\tan^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{x}{y}\frac{\left( 1 - \tan a \right)}{\left( 1 + \tan a \right)}\] ?
Differentiate \[x^{\sin x}\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\cot x} + \left( \cot x \right)^{\tan x}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[\left( \cos x \right)^y = \left( \tan y \right)^x\] , prove that \[\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
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 ______________ .
Find the second order derivatives of the following function log (sin x) ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If `x = sin(1/2 log y)` show that (1 − x2)y2 − xy1 − a2y = 0.
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If y = xx, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
