Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 e^{x^{2}} e^{y^{2}} + 5 \log{\left(y \right)} \cos{\left(x^{3} \right)}=-22

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 15 x^{2} \log{\left(y \right)} \sin{\left(x^{3} \right)} - 18 x e^{x^{2}} e^{y^{2}} - 18 y y' e^{x^{2}} e^{y^{2}} + \frac{5 y' \cos{\left(x^{3} \right)}}{y} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{3 x y \left(5 x \log{\left(y \right)} \sin{\left(x^{3} \right)} + 6 e^{x^{2} + y^{2}}\right)}{18 y^{2} e^{x^{2} + y^{2}} - 5 \cos{\left(x^{3} \right)}}