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

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