Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{x} e^{y^{2}} + 2 y^{2} \log{\left(x \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 36 \sqrt{x} y y' e^{y^{2}} + 4 y y' \log{\left(x \right)} + \frac{2 y^{2}}{x} - \frac{9 e^{y^{2}}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 2 \sqrt{x} y^{2} + 9 x e^{y^{2}}}{4 y \left(x^{\frac{3}{2}} \log{\left(x \right)} - 9 x^{2} e^{y^{2}}\right)}