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)} + 3 y \log{\left(x \right)}=34

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{x y'}{y} + 3 y' \log{\left(x \right)} - \log{\left(y \right)} + \frac{3 y}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{y \left(- x \log{\left(y \right)} + 3 y\right)}{x \left(x - 3 y \log{\left(x \right)}\right)}