Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 9 x^{2} e^{y^{2}} - 4 y^{2} \cos{\left(x \right)}=23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 18 x^{2} y y' e^{y^{2}} - 18 x e^{y^{2}} + 4 y^{2} \sin{\left(x \right)} - 8 y y' \cos{\left(x \right)} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{- 9 x e^{y^{2}} + 2 y^{2} \sin{\left(x \right)}}{y \left(9 x^{2} e^{y^{2}} + 4 \cos{\left(x \right)}\right)}$