Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 30 \sqrt{x} \sqrt{y} + 2 \cos{\left(x^{3} \right)} \cos{\left(y^{3} \right)}=-36

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{15 \sqrt{x} y'}{\sqrt{y}} - 6 x^{2} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} - 6 y^{2} y' \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} + \frac{15 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{- 2 x^{\frac{5}{2}} \sqrt{y} \sin{\left(x^{3} \right)} \cos{\left(y^{3} \right)} + 5 y}{2 \sqrt{x} y^{\frac{5}{2}} \sin{\left(y^{3} \right)} \cos{\left(x^{3} \right)} - 5 x}