Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle - 7 \sqrt{3} \sqrt{y} e^{x^{3}} - 3 \sin{\left(x \right)} \cos{\left(y^{3} \right)}=14$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 21 \sqrt{3} x^{2} \sqrt{y} e^{x^{3}} + 9 y^{2} y' \sin{\left(x \right)} \sin{\left(y^{3} \right)} - 3 \cos{\left(x \right)} \cos{\left(y^{3} \right)} - \frac{7 \sqrt{3} y' e^{x^{3}}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{6 \left(7 \sqrt{3} x^{2} y e^{x^{3}} + \sqrt{y} \cos{\left(x \right)} \cos{\left(y^{3} \right)}\right)}{18 y^{\frac{5}{2}} \sin{\left(x \right)} \sin{\left(y^{3} \right)} - 7 \sqrt{3} e^{x^{3}}}$