Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 x^{3} \log{\left(y \right)} - 12 \sqrt{y} e^{x^{2}}=12$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: \frac{3 x^{3} y'}{y} + 9 x^{2} \log{\left(y \right)} - 24 x \sqrt{y} e^{x^{2}} - \frac{6 y' e^{x^{2}}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{x \left(- 3 x y^{\frac{3}{2}} \log{\left(y \right)} + 8 y^{2} e^{x^{2}}\right)}{x^{3} \sqrt{y} - 2 y e^{x^{2}}}$