Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 2 \sqrt{5} x^{3} \sqrt{y} - 6 e^{x} \log{\left(y \right)}=-3$

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