Question

find dy/dx, if y=x^logx​

Answer 1

Given :

$y = {{x}^{log \: x} }$

To find :

$\dfrac{dy}{dx}$

Solution :

$\implies \: y = {{x}^{log \: x} }$

Taking log both sides :-

$\implies \:log \: y =log ( {{x}^{log \: x} })$

$\implies \:log \: y =log (x )\: \: log ( {x})$

$\implies \:log \: y = ({log \: {x})}^{2}$

Now differentiating both sides :-

$\implies \: \dfrac{d(log \: y)}{dx} =\dfrac{d(({log \: {x})}^{2} )}{dx}$

using Chain rule :-

$\implies \dfrac{1}{y} \dfrac{dy}{dx} =2log (x ) \times \dfrac{d(log (x ))}{dx} \times \dfrac{dx}{dx}$

We know that :-

d(log t)/dt = 1/t

$\implies \dfrac{1}{y} \dfrac{dy}{dx} =2log (x ) \times \dfrac{1}{x} \times 1$

$\implies \dfrac{1}{y} \dfrac{dy}{dx} = \dfrac{2log (x )}{x}$

$\implies \dfrac{dy}{dx} = \dfrac{2log (x )}{x} \times y$

Now put :-

$y = {{x}^{log \: x} }$

$\implies \dfrac{dy}{dx} = \dfrac{2log (x )}{x} \times {{x}^{log \: x} }$

$\implies \dfrac{dy}{dx} = (\dfrac{2log (x )}{x} ) {{x}^{log \: x} }$

Answer 2

Given ,

The function is $\tt y = {x}^{log(x)}$

Taking log on both sides , we get

$\tt \implies log(y) = log ({x}^{ log(x) } )$

$\tt \implies log(y) =log(x) \times log ({x}^{ } )$

$\tt \implies log(y) = \{{log(x) \}}^{2}$

Differentiating wrt x by using chain rule , we get

$\tt \implies \frac{d}{dx} log(y) = \frac{d}{dx} \{{log(x) \}}^{2}$

$\tt \implies \frac{1}{y} \frac{dy}{dx} = 2log(x) \frac{d}{dx} log(x)$

$\tt \implies \frac{1}{y} \frac{dy}{dx} = 2log(x) \times \frac{1}{x}$

$\tt \implies \frac{1}{y} \frac{dy}{dx} = \frac{2log(x) }{x}$

$\tt \implies \frac{dy}{dx} = y \{\frac{2log(x) }{x} \}$

$\tt \implies \frac{dy}{dx} = {x}^{log(x)} \{\frac{2log(x) }{x} \}$

$\tt \implies \frac{dy}{dx} = 2{x}^{log(x) - 1} \times log(x)$

Remmember :

$\tt \mapsto \frac{d}{dx} log(x) = \frac{1}{x}$

$\tt \mapsto log( {m}^{n} ) = n \times log(m)$