Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 6 \sqrt{y} \cos{\left(x^{3} \right)} - 2 e^{x^{3}} e^{y^{3}}=22

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 18 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} - 6 x^{2} e^{x^{3}} e^{y^{3}} - 6 y^{2} y' e^{x^{3}} e^{y^{3}} + \frac{3 y' \cos{\left(x^{3} \right)}}{\sqrt{y}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 x^{2} \left(\sqrt{y} e^{x^{3} + y^{3}} + 3 y \sin{\left(x^{3} \right)}\right)}{- 2 y^{\frac{5}{2}} e^{x^{3} + y^{3}} + \cos{\left(x^{3} \right)}}