Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 3 \sqrt{7} \sqrt{x} e^{y} + 5 y \sin{\left(x^{3} \right)}=-33$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 3 \sqrt{7} \sqrt{x} y' e^{y} + 15 x^{2} y \cos{\left(x^{3} \right)} + 5 y' \sin{\left(x^{3} \right)} + \frac{3 \sqrt{7} e^{y}}{2 \sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{30 x^{\frac{5}{2}} y \cos{\left(x^{3} \right)} + 3 \sqrt{7} e^{y}}{10 \sqrt{x} \sin{\left(x^{3} \right)} + 6 \sqrt{7} x e^{y}}$