Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 8 \sqrt{x} e^{y} - 3 \log{\left(y \right)} \sin{\left(x^{3} \right)}=2$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 8 \sqrt{x} y' e^{y} - 9 x^{2} \log{\left(y \right)} \cos{\left(x^{3} \right)} - \frac{3 y' \sin{\left(x^{3} \right)}}{y} + \frac{4 e^{y}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{y \left(- 9 x^{\frac{5}{2}} \log{\left(y \right)} \cos{\left(x^{3} \right)} + 4 e^{y}\right)}{3 \sqrt{x} \sin{\left(x^{3} \right)} - 8 x y e^{y}}$