Question

what is the derivative of y=x^e^x​

Answer 1

Step-by-step explanation:

the given function is

$y = {x}^{ {e}^{x} }$

taking log on both the sides

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

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

differentiation wrt x

$\frac{1}{y} \times \frac{dy}{dx} = {e}^{x} \times \frac{1}{x} + logx \: {e}^{x}$

$\frac{dy}{dx} = y( {e}^{x} \times \frac{1}{x} + log \: x \times {e}^{x} )$

$\frac{dy}{dx} = y( {e}^{x} ( \frac{1}{x} + logx))$

$put \: y \: = {x}^{ {e}^{x} }$

$\frac{dy}{dx} = {x}^{ {e}^{x} } ( {e}^{x} ( \frac{1}{x} + log \: x))$