Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 6 \sqrt{2} \sqrt{x} e^{y^{2}} + 5 \log{\left(x \right)} \cos{\left(y \right)}=-23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 12 \sqrt{2} \sqrt{x} y y' e^{y^{2}} - 5 y' \log{\left(x \right)} \sin{\left(y \right)} + \frac{5 \cos{\left(y \right)}}{x} - \frac{3 \sqrt{2} e^{y^{2}}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{5 \sqrt{x} \cos{\left(y \right)} - 3 \sqrt{2} x e^{y^{2}}}{5 x^{\frac{3}{2}} \log{\left(x \right)} \sin{\left(y \right)} + 12 \sqrt{2} x^{2} y e^{y^{2}}}$