Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 3 x^{2} e^{y} + 5 \sqrt{7} \sqrt{y} \cos{\left(x \right)}=-23$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 3 x^{2} y' e^{y} - 6 x e^{y} - 5 \sqrt{7} \sqrt{y} \sin{\left(x \right)} + \frac{5 \sqrt{7} y' \cos{\left(x \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{12 x \sqrt{y} e^{y} + 10 \sqrt{7} y \sin{\left(x \right)}}{6 x^{2} \sqrt{y} e^{y} - 5 \sqrt{7} \cos{\left(x \right)}}$