Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 9 \sqrt{x} e^{y^{2}} + 2 \sqrt{y} \cos{\left(x^{3} \right)}=20

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 18 \sqrt{x} y y' e^{y^{2}} - 6 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + \frac{y' \cos{\left(x^{3} \right)}}{\sqrt{y}} + \frac{9 e^{y^{2}}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{3 \left(4 x^{\frac{5}{2}} y \sin{\left(x^{3} \right)} - 3 \sqrt{y} e^{y^{2}}\right)}{2 \left(\sqrt{x} \cos{\left(x^{3} \right)} + 18 x y^{\frac{3}{2}} e^{y^{2}}\right)}