Answer to Question #320055 in Calculus for Laina

Question

Find y' if x^y^2=y^x^2

Answer 1

$x^{y^2}=y^{x^2}\\y^2\ln x=x^2\ln y\\d\left( y^2\ln x \right) =d\left( x^2\ln y \right) \\2y\ln xdy+\frac{y^2}{x}dx=2x\ln ydx+\frac{x^2}{y}dy\\\left( 2y\ln x-\frac{x^2}{y} \right) dy=\left( 2x\ln y-\frac{y^2}{x} \right) dx\\\frac{dy}{dx}=\frac{2x\ln y-\frac{y^2}{x}}{2y\ln x-\frac{x^2}{y}}$