Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle - 15 \sqrt{x} e^{y^{2}} - 6 e^{x^{2}} \cos{\left(y \right)}=45

Taking the derivative of both sides using implicit differentiation gives: LaTeX: - 30 \sqrt{x} y y' e^{y^{2}} - 12 x e^{x^{2}} \cos{\left(y \right)} + 6 y' e^{x^{2}} \sin{\left(y \right)} - \frac{15 e^{y^{2}}}{2 \sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = \frac{8 x^{\frac{3}{2}} e^{x^{2}} \cos{\left(y \right)} + 5 e^{y^{2}}}{4 \left(\sqrt{x} e^{x^{2}} \sin{\left(y \right)} - 5 x y e^{y^{2}}\right)}