Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 18 \sqrt{2} \sqrt{x} e^{y^{2}} + 9 \sin{\left(y^{3} \right)} \cos{\left(x^{2} \right)}=2

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 36 \sqrt{2} \sqrt{x} y y' e^{y^{2}} - 18 x \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + 27 y^{2} y' \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - \frac{9 \sqrt{2} e^{y^{2}}}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{2 x^{\frac{3}{2}} \sin{\left(x^{2} \right)} \sin{\left(y^{3} \right)} + \sqrt{2} e^{y^{2}}}{y \left(3 \sqrt{x} y \cos{\left(x^{2} \right)} \cos{\left(y^{3} \right)} - 4 \sqrt{2} x e^{y^{2}}\right)}