Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 9 \sqrt{3} \sqrt{x} \cos{\left(y^{3} \right)} + 8 \sqrt{3} \sqrt{y} \cos{\left(x^{3} \right)}=10

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 27 \sqrt{3} \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} - 24 \sqrt{3} x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + \frac{4 \sqrt{3} y' \cos{\left(x^{3} \right)}}{\sqrt{y}} - \frac{9 \sqrt{3} \cos{\left(y^{3} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 \left(16 x^{\frac{5}{2}} y \sin{\left(x^{3} \right)} + 3 \sqrt{y} \cos{\left(y^{3} \right)}\right)}{2 \left(4 \sqrt{x} \cos{\left(x^{3} \right)} + 27 x y^{\frac{5}{2}} \sin{\left(y^{3} \right)}\right)}