Advertisements
Advertisements
Question
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
Advertisements
Solution
\[\text{ Let y }= x^{\log x }+ \left( \log x \right)^x \]
\[\text{ Also, let u } = \left( \log x \right)^x \text{ and v} = x^{\log x} \]
\[ \therefore y = v + u\]
\[ \Rightarrow \frac{dy}{dx} = \frac{dv}{dx} + \frac{du}{dx} . . . \left( i \right)\]
\[\text{ Now, u} = \left( \log x \right)^x \]
\[ \Rightarrow \log u = \log\left[ \left( \log x \right)^x \right]\]
\[ \Rightarrow \log u = x\log\left( \log x \right)\]
Differentiating both sides with respect to x,
\[\frac{1}{u}\frac{du}{dx} = \log\left( \log x \right)\frac{d}{dx}\left( x \right) + x\frac{d}{dx}\left[ \log\left( \log x \right) \right]\]
\[ \Rightarrow \frac{du}{dx} = u\left[ \log\left( \log x \right) + x\frac{1}{\log x}\frac{d}{dx}\left( \log x \right) \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( \log x \right)^x \left[ \log\left( \log x \right) + \frac{x}{\log x} \times \frac{1}{x} \right]\]
\[ \Rightarrow \frac{du}{dx} = \left( \log x \right)^x \left[ \log\left( \log x \right) + \frac{1}{\log x} \right] . . . \left( ii \right)\]
\[\text{ Also, v} = x^{\log x} \]
\[ \Rightarrow \log v = \log x^{\log x} \]
\[ \Rightarrow \log v = \log x \log x = \left( \log x \right)^2 \]
Differentiating both sides with respect to x,
\[\frac{1}{v}\frac{dv}{dx} = \frac{d}{dx}\left[ \left( \log x \right)^2 \right]\]
\[ \Rightarrow \frac{1}{v}\frac{dv}{dx} = 2\left( \log x \right)\frac{d}{dx}\left( \log x \right)\]
\[ \Rightarrow \frac{dv}{dx} = 2v\left( \log x \right)\frac{1}{x}\]
\[ \Rightarrow \frac{dv}{dx} = 2 x^{\log x} \frac{\log x}{x}\]
\[ \Rightarrow \frac{dv}{dx} = 2 x^{\log x} \frac{\log x}{x} . . . \left( iii \right)\]
\[\text{ From} \left( i \right), \left( ii \right) \text{ and }\left( iii \right), \text{ we obtain}\]
\[\frac{dy}{dx} = 2 x^{\log x} \frac{\log x}{x} + \left( \log x \right)^x \left[ \log\left( \log x \right) + \frac{1}{\log x} \right]\]
APPEARS IN
RELATED QUESTIONS
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate etan x ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
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 \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
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 \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
Find the second order derivatives of the following function x cos x ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]
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 = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
f(x) = 3x2 + 6x + 8, x ∈ R
