Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 2 \sqrt{3} \sqrt{x} \log{\left(y \right)} - 6 \sqrt{3} \sqrt{y} e^{x^{2}}=1

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