Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{6} \sqrt{x} \log{\left(y \right)} - 4 x^{3} e^{y^{2}}=9

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{3 \sqrt{6} \sqrt{x} y'}{y} - 8 x^{3} y y' e^{y^{2}} - 12 x^{2} e^{y^{2}} + \frac{3 \sqrt{6} \log{\left(y \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 y \left(- 8 x^{\frac{5}{2}} e^{y^{2}} + \sqrt{6} \log{\left(y \right)}\right)}{2 \left(8 x^{\frac{7}{2}} y^{2} e^{y^{2}} - 3 \sqrt{6} x\right)}