Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{x} \cos{\left(y^{3} \right)} - 2 y^{2} \cos{\left(x^{3} \right)}=-50

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} + 6 x^{2} y^{2} \sin{\left(x^{3} \right)} - 4 y y' \cos{\left(x^{3} \right)} - \frac{2 \cos{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 3 x^{\frac{5}{2}} y^{2} \sin{\left(x^{3} \right)} + \cos{\left(y^{3} \right)}}{2 y \left(- \sqrt{x} \cos{\left(x^{3} \right)} + 3 x y \sin{\left(y^{3} \right)}\right)}