Question

Find $\frac{dy}{dx}$, if $x^y+y^x=b^a+a^b$

Answer 1

$x^y+y^x=Constant$
Differentiating with respect to x,

$x^y(\frac{dy}{dx}lnx+\frac{y}{x})+y^x(lny+\frac{x}{y}\frac{dy}{dx})=0$

$x^y{y}^{\prime }lnx+x^{y-1}y+y^{x}lny+y^{x-1}{y}^{\prime }=0$

$y^{\prime }(x^{y}lnx+y^{x-1})=-(x^{y-1}y+y^{x}lny)$

$y^{\prime }=\frac{-(x^{y-1}y+y^{x}lny)}{(x^{y}lnx+y^{x-1})}$