Treat y as a function of x and use the Chain Rule whenever y appears. See here for more details on implicit differentiation.
Here, differentiating both sides with respect to x and taking the Product Rule and Chain Rule into account yields
\frac{dy}{dx}\cos(x^2)-2xy\sin(x^2)-2y\frac{dy}{dx}=y+x\frac{dy}{dx}-2x
Put all terms that contain \frac{dy}{dx} on one side, all the other terms on the other sides:
\frac{dy}{dx}\cos(x^2)-2y\frac{dy}{dx}-x\frac{dy}{dx}=2xy\sin(x^2)+y-2x
Factoring \frac{dy}{dx} and dividing, we obtain
\frac{dy}{dx}=\frac{2xy\sin(x^2)+y-2x}{\cos(x^2)-2y-x}
