Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{6} \sqrt{y} \sin{\left(x^{3} \right)} + 5 y^{2} e^{x^{2}}=-32$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 18 \sqrt{6} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + 10 x y^{2} e^{x^{2}} + 10 y y' e^{x^{2}} - \frac{3 \sqrt{6} y' \sin{\left(x^{3} \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 x \left(9 \sqrt{6} x y \cos{\left(x^{3} \right)} - 5 y^{\frac{5}{2}} e^{x^{2}}\right)}{10 y^{\frac{3}{2}} e^{x^{2}} - 3 \sqrt{6} \sin{\left(x^{3} \right)}}$