Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 4 \sqrt{y} \cos{\left(x \right)} + 3 y^{3} e^{x^{2}}=-14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 6 x y^{3} e^{x^{2}} + 4 \sqrt{y} \sin{\left(x \right)} + 9 y^{2} y' e^{x^{2}} - \frac{2 y' \cos{\left(x \right)}}{\sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 \left(3 x y^{\frac{7}{2}} e^{x^{2}} + 2 y \sin{\left(x \right)}\right)}{- 9 y^{\frac{5}{2}} e^{x^{2}} + 2 \cos{\left(x \right)}}$