Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 y^{3} \cos{\left(x \right)} + 2 e^{x^{2}} \cos{\left(y \right)}=-18$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 4 x e^{x^{2}} \cos{\left(y \right)} + 8 y^{3} \sin{\left(x \right)} - 24 y^{2} y' \cos{\left(x \right)} - 2 y' e^{x^{2}} \sin{\left(y \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(x e^{x^{2}} \cos{\left(y \right)} + 2 y^{3} \sin{\left(x \right)}\right)}{12 y^{2} \cos{\left(x \right)} + e^{x^{2}} \sin{\left(y \right)}}$