Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 e^{x} \log{\left(y \right)} + 3 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=-3$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 6 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 9 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 8 e^{x} \log{\left(y \right)} - \frac{8 y' e^{x}}{y} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 y \left(3 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 4 e^{x} \log{\left(y \right)}\right)}{9 y^{3} \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 8 e^{x}}$