Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 3 x \cos{\left(y^{2} \right)} - 8 \sin{\left(y^{2} \right)} \cos{\left(x^{2} \right)}=-36

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 6 x y y' \sin{\left(y^{2} \right)} + 16 x \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} - 16 y y' \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)} - 3 \cos{\left(y^{2} \right)} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 16 x \sin{\left(x^{2} \right)} \sin{\left(y^{2} \right)} + 3 \cos{\left(y^{2} \right)}}{2 y \left(3 x \sin{\left(y^{2} \right)} - 8 \cos{\left(x^{2} \right)} \cos{\left(y^{2} \right)}\right)}