Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 5 y \log{\left(x \right)} + 2 e^{x} \cos{\left(y^{3} \right)}=-21

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 6 y^{2} y' e^{x} \sin{\left(y^{3} \right)} - 5 y' \log{\left(x \right)} + 2 e^{x} \cos{\left(y^{3} \right)} - \frac{5 y}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 x e^{x} \cos{\left(y^{3} \right)} - 5 y}{x \left(6 y^{2} e^{x} \sin{\left(y^{3} \right)} + 5 \log{\left(x \right)}\right)}