Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 2 x^{2} e^{y} - 14 \sqrt{y} \cos{\left(x \right)}=32$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 2 x^{2} y' e^{y} + 4 x e^{y} + 14 \sqrt{y} \sin{\left(x \right)} - \frac{7 y' \cos{\left(x \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{4 x \sqrt{y} e^{y} + 14 y \sin{\left(x \right)}}{2 x^{2} \sqrt{y} e^{y} - 7 \cos{\left(x \right)}}$