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

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