Question

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

Answer 1

$x^{y²}=y^{x²}$

Take natural log of both sides

$x²lny=y²lnx$

Differentiate both sides with respect to x

$2xlny+\frac{x²}{y}y'=(2ylnx)y'+\frac{y²}{x}$

Multiply through by $xy$

$2x²ylny+{x³}y'=(2y²xlnx)y'+y³$

$=> {x³}y'-(2y²xlnx)y'=y³-2x²ylny$

$=> (x³-2y²xlnx)y'=y³-2x²ylny$

$=> y'=\frac{y³-2x²ylny}{x³-2y²xlnx}$