Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - y^{3} \log{\left(x \right)} - 4 e^{y} \cos{\left(x^{2} \right)}=36

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 8 x e^{y} \sin{\left(x^{2} \right)} - 3 y^{2} y' \log{\left(x \right)} - 4 y' e^{y} \cos{\left(x^{2} \right)} - \frac{y^{3}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{2} e^{y} \sin{\left(x^{2} \right)} - y^{3}}{x \left(3 y^{2} \log{\left(x \right)} + 4 e^{y} \cos{\left(x^{2} \right)}\right)}