Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 9 \sqrt{7} \sqrt{y} \sin{\left(x^{3} \right)} - \cos{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=5$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: 27 \sqrt{7} x^{2} \sqrt{y} \cos{\left(x^{3} \right)} + 3 x^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 3 y^{2} y' \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + \frac{9 \sqrt{7} y' \sin{\left(x^{3} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = - \frac{2 x^{2} \left(\sqrt{y} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 9 \sqrt{7} y \cos{\left(x^{3} \right)}\right)}{2 y^{\frac{5}{2}} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + 3 \sqrt{7} \sin{\left(x^{3} \right)}}$