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

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 8 x^{3} y y' e^{y^{2}} + 12 x^{2} e^{y^{2}} - 2 \sqrt{3} x \sqrt{y} \sin{\left(x^{2} \right)} + \frac{\sqrt{3} y' \cos{\left(x^{2} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{4 x \left(- 6 x \sqrt{y} e^{y^{2}} + \sqrt{3} y \sin{\left(x^{2} \right)}\right)}{16 x^{3} y^{\frac{3}{2}} e^{y^{2}} + \sqrt{3} \cos{\left(x^{2} \right)}}$