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

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 42 \sqrt{x} y^{2} y' \cos{\left(y^{3} \right)} + 42 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} - \frac{7 y' \cos{\left(x^{3} \right)}}{\sqrt{y}} - \frac{7 \sin{\left(y^{3} \right)}}{\sqrt{x}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{6 x^{\frac{5}{2}} y \sin{\left(x^{3} \right)} - \sqrt{y} \sin{\left(y^{3} \right)}}{\sqrt{x} \cos{\left(x^{3} \right)} + 6 x y^{\frac{5}{2}} \cos{\left(y^{3} \right)}}$