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}} - 3 \sqrt{7} \sqrt{y} \log{\left(x \right)}=-39

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 x^{2} y y' e^{y^{2}} + 18 x e^{y^{2}} - \frac{3 \sqrt{7} y' \log{\left(x \right)}}{2 \sqrt{y}} - \frac{3 \sqrt{7} \sqrt{y}}{x} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 \left(- 6 x^{2} \sqrt{y} e^{y^{2}} + \sqrt{7} y\right)}{x \left(12 x^{2} y^{\frac{3}{2}} e^{y^{2}} - \sqrt{7} \log{\left(x \right)}\right)}