Question

Find the derivative of x^logx wet logx.

Answer 1

Answer:

$\dfrac{d( \: {x}^{ logx }) }{d( \: logx )} = \: 2 \: {x}^{ log(x) } \: logx$

Step-by-step explanation:

To find ----->

$derivative \: of \: {x}^{ logx } \: with \: respect \: to \: logx$

Concept used ---->

1)

$log( {x}^{m} ) = \: m \: logx$

2)

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

3)

$\frac{d}{dx} ( logx ) \: = \frac{1}{x}$

4)

$if \: u \: and \: v \: are \: two \: functions \: of \\ \: x \: and \: we \: have \: to \: find \: derivtive \\ of \: u \: with \: respect \: to \: v \: then \\ \dfrac{du}{dv } = \dfrac{ \dfrac{du}{dx} }{ \dfrac{dv}{dx} }$

Solution---->

$let \: u \: = {x}^{ logx } \: and \: v = logx$

$now \: u \: = {x}^{ logx }$

$taking \: log \: both \: sides \: we \: get$

$logu = log( {x}^{ logx } )$

$\: \: \: = logx \: logx$

$\: \: \: = {( logx) }^{2}$

$differentiating \: with \: respect \: to \: x \: we \: get$

$\dfrac{d}{dx} ( \: logu) \: = \frac{d}{dx} {( logx) }^{2}$

$\dfrac{1}{u} \dfrac{du}{dx} \: = 2 { (logx) }^{2 - 1} \frac{d}{dx} \: ( logx)$

$\dfrac{du}{dx} = u \: 2 \: logx ( \frac{1}{x} )$

$\dfrac{du}{dx} \: = \dfrac{2 \: {x}^{ logx } \: logx }{x}$

$now \: v \: = logx$

$differentiating \: with \: respect \: to \\ \: x \: we \: get$

$\dfrac{dv}{dx} = \dfrac{d}{dx} ( logx)$

$\dfrac{dv}{dx} = \dfrac{d}{dx} ( logx)$

$\dfrac{dv}{dx} = \dfrac{1}{x}$

$now \: derivative \: of \: {x}^{ logx } with \\ \: respect \: to \: logx \: is$

$\dfrac{du}{dv} \: = \dfrac{ \dfrac{du}{dx} }{ \dfrac{dv}{dx} }$

$\dfrac{d ({x}^{ logx }) }{d( logx) } = \dfrac{ \dfrac{2 \: {x}^{logx} \: logx }{x} }{ \dfrac{1}{x} }$

= 2 xˡᵒᵍˣ logx