Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle x \log{\left(y \right)} + 8 y \log{\left(x \right)}=32

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{x y'}{y} + 8 y' \log{\left(x \right)} + \log{\left(y \right)} + \frac{8 y}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{y \left(x \log{\left(y \right)} + 8 y\right)}{x \left(x + 8 y \log{\left(x \right)}\right)}