Use implicit differentiation where LaTeX: \displaystyle y is a function of LaTeX: \displaystyle x to find the derivative of LaTeX: \displaystyle 12 \sqrt{x} y + 9 x e^{y^{2}}=4

Taking the derivative of both sides using implicit differentiation gives: LaTeX: 12 \sqrt{x} y' + 18 x y y' e^{y^{2}} + 9 e^{y^{2}} + \frac{6 y}{\sqrt{x}} = 0 Solving for LaTeX: \displaystyle y' gives LaTeX: \displaystyle y' = - \frac{3 \sqrt{x} e^{y^{2}} + 2 y}{6 x^{\frac{3}{2}} y e^{y^{2}} + 4 x}