Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 7 \sqrt{2} \sqrt{y} \cos{\left(x^{2} \right)} - 5 y^{2} e^{x^{3}}=-20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 15 x^{2} y^{2} e^{x^{3}} + 14 \sqrt{2} x \sqrt{y} \sin{\left(x^{2} \right)} - 10 y y' e^{x^{3}} - \frac{7 \sqrt{2} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 x \left(- 15 x y^{\frac{5}{2}} e^{x^{3}} + 14 \sqrt{2} y \sin{\left(x^{2} \right)}\right)}{20 y^{\frac{3}{2}} e^{x^{3}} + 7 \sqrt{2} \cos{\left(x^{2} \right)}}