Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle x^{3} \cos{\left(y^{2} \right)} - 2 y^{3} \cos{\left(x^{2} \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 2 x^{3} y y' \sin{\left(y^{2} \right)} + 3 x^{2} \cos{\left(y^{2} \right)} + 4 x y^{3} \sin{\left(x^{2} \right)} - 6 y^{2} y' \cos{\left(x^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{x \left(3 x \cos{\left(y^{2} \right)} + 4 y^{3} \sin{\left(x^{2} \right)}\right)}{2 y \left(x^{3} \sin{\left(y^{2} \right)} + 3 y \cos{\left(x^{2} \right)}\right)}