Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 e^{y} \cos{\left(x \right)} - \log{\left(x \right)} \cos{\left(y \right)}=32

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 5 y' e^{y} \cos{\left(x \right)} + y' \log{\left(x \right)} \sin{\left(y \right)} + 5 e^{y} \sin{\left(x \right)} - \frac{\cos{\left(y \right)}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{5 x e^{y} \sin{\left(x \right)} - \cos{\left(y \right)}}{x \left(5 e^{y} \cos{\left(x \right)} - \log{\left(x \right)} \sin{\left(y \right)}\right)}