Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 16 \sqrt{x} y - 4 e^{x^{3}} \cos{\left(y \right)}=37

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 16 \sqrt{x} y' - 12 x^{2} e^{x^{3}} \cos{\left(y \right)} + 4 y' e^{x^{3}} \sin{\left(y \right)} - \frac{8 y}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 x^{\frac{5}{2}} e^{x^{3}} \cos{\left(y \right)} + 2 y}{\sqrt{x} e^{x^{3}} \sin{\left(y \right)} - 4 x}