Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 8 y^{3} e^{x^{2}} + 7 e^{y^{2}} \log{\left(x \right)}=-13

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 16 x y^{3} e^{x^{2}} - 24 y^{2} y' e^{x^{2}} + 14 y y' e^{y^{2}} \log{\left(x \right)} + \frac{7 e^{y^{2}}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 16 x^{2} y^{3} e^{x^{2}} + 7 e^{y^{2}}}{2 x y \left(12 y e^{x^{2}} - 7 e^{y^{2}} \log{\left(x \right)}\right)}