Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 36 \sqrt{x} \sqrt{y} + 8 y^{2} \cos{\left(x^{3} \right)}=33

Taking the derivative of both sides using implicit differentiation gives: LaTeX: \frac{18 \sqrt{x} y'}{\sqrt{y}} - 24 x^{2} y^{2} \sin{\left(x^{3} \right)} + 16 y y' \cos{\left(x^{3} \right)} + \frac{18 \sqrt{y}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 \left(4 x^{\frac{5}{2}} y^{\frac{5}{2}} \sin{\left(x^{3} \right)} - 3 y\right)}{8 \sqrt{x} y^{\frac{3}{2}} \cos{\left(x^{3} \right)} + 9 x}