Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - x^{2} \log{\left(y \right)} + 3 \sqrt{2} \sqrt{y} e^{x}=47$

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