Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 3 \sqrt{3} \sqrt{x} \cos{\left(y^{3} \right)} + 5 x \sin{\left(y \right)}=-22

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 9 \sqrt{3} \sqrt{x} y^{2} y' \sin{\left(y^{3} \right)} + 5 x y' \cos{\left(y \right)} + 5 \sin{\left(y \right)} + \frac{3 \sqrt{3} \cos{\left(y^{3} \right)}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{10 \sqrt{x} \sin{\left(y \right)} + 3 \sqrt{3} \cos{\left(y^{3} \right)}}{2 \left(- 5 x^{\frac{3}{2}} \cos{\left(y \right)} + 9 \sqrt{3} x y^{2} \sin{\left(y^{3} \right)}\right)}