Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle \sqrt{2} \sqrt{x} e^{y} + \log{\left(y \right)} \sin{\left(x^{3} \right)}=-18$

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