Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 14 \sqrt{x} \sin{\left(y^{3} \right)} - 5 e^{x^{2}} e^{y^{2}}=-16

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 42 \sqrt{x} y^{2} y' \cos{\left(y^{3} \right)} - 10 x e^{x^{2}} e^{y^{2}} - 10 y y' e^{x^{2}} e^{y^{2}} - \frac{7 \sin{\left(y^{3} \right)}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{10 x^{\frac{3}{2}} e^{x^{2} + y^{2}} + 7 \sin{\left(y^{3} \right)}}{2 y \left(5 \sqrt{x} e^{x^{2} + y^{2}} + 21 x y \cos{\left(y^{3} \right)}\right)}