Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 4 \sqrt{3} \sqrt{y} \cos{\left(x \right)} - 5 y^{2} e^{x^{3}}=-13

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 15 x^{2} y^{2} e^{x^{3}} + 4 \sqrt{3} \sqrt{y} \sin{\left(x \right)} - 10 y y' e^{x^{3}} - \frac{2 \sqrt{3} y' \cos{\left(x \right)}}{\sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 15 x^{2} y^{\frac{5}{2}} e^{x^{3}} + 4 \sqrt{3} y \sin{\left(x \right)}}{2 \left(5 y^{\frac{3}{2}} e^{x^{3}} + \sqrt{3} \cos{\left(x \right)}\right)}