Question

differentiate x^y= e^x . with respect to x

Answer 1

$x^y=e^x$

taking ln on both sides

y × ln x = x 

y = x / ln x

dy/dx = {ln x dx - x d (ln x)}/(ln x)²

dy/dx = {ln x - 1}/(ln x)²

$\frac{dy}{dx}=\frac{lnx-1}{(lnx)^2}$

Answer 2

$x^y = e^x\\\\y*Ln\ x=x*Ln\ e=x\\\\y=\frac{x}{Ln\ x}\\\\\frac{dy}{dx}=x*\frac{-1}{(Ln\ x)^2}*(1/x)+\frac{1}{Ln\ x}*1\\\\=\frac{-1+Ln\ x}{(Ln\ x)^2},\ \ \ -- answer\ 1\\\\we\ know,\ \ Ln\ x=\frac{x}{y},\ substuting\ it,\\\\\frac{dy}{dx}=\frac{-1+(x/y)}{x^2/y^2}=\frac{(x-y)y}{x^2},\ \ \ -- answer\ 2$