Answer
$$\frac{dy}{dx}=\frac{2-2xye^{x^2y}}{x^2e^{x^2y}-2}$$
Work Step by Step
$$e^{x^2y}=2x+2y$$
1) Differentiate both sides of the equation with respect to $x$:
$$\frac{d}{dx}(e^{x^2y})=\frac{d}{dx}(2x+2y)$$
$$e^{x^2y}\frac{d}{dx}(x^2y)=2+2\frac{dy}{dx}$$
$$e^{x^2y}(2xy+x^2\frac{dy}{dx})=2+2\frac{dy}{dx}$$
$$2xye^{x^2y}+x^2e^{x^2y}\frac{dy}{dx}=2+2\frac{dy}{dx}$$
2) Collect all the terms with $dy/dx$ onto one side and solve for $dy/dx$:
$$x^2e^{x^2y}\frac{dy}{dx}-2\frac{dy}{dx}=2-2xye^{x^2y}$$
$$\frac{dy}{dx}=\frac{2-2xye^{x^2y}}{x^2e^{x^2y}-2}$$
