Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 8 \sqrt{x} e^{y} - 12 \sqrt{y} \log{\left(x \right)}=13$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 8 \sqrt{x} y' e^{y} - \frac{6 y' \log{\left(x \right)}}{\sqrt{y}} - \frac{12 \sqrt{y}}{x} - \frac{4 e^{y}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{6 \sqrt{x} y + 2 x \sqrt{y} e^{y}}{3 x^{\frac{3}{2}} \log{\left(x \right)} + 4 x^{2} \sqrt{y} e^{y}}$