Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 35 \sqrt{x} \sqrt{y} + 9 e^{y} \sin{\left(x^{3} \right)}=-13

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - \frac{35 \sqrt{x} y'}{2 \sqrt{y}} + 27 x^{2} e^{y} \cos{\left(x^{3} \right)} + 9 y' e^{y} \sin{\left(x^{3} \right)} - \frac{35 \sqrt{y}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 54 x^{\frac{5}{2}} \sqrt{y} e^{y} \cos{\left(x^{3} \right)} + 35 y}{18 \sqrt{x} \sqrt{y} e^{y} \sin{\left(x^{3} \right)} - 35 x}