Finding the derivative of a node What are the slopes of the curve $y^2=x^3+x^2$ at the origin.
The graph indicates it passes through the origin twice with slopes $1$ and $-1$. But how could I verify this using calculus?
Edit: (Not just verify, but derive these numbers without using the graph)

Note that $y=\pm x\sqrt{1+x}$. Therefore, $y/x = \pm\sqrt{1+x}$. But that means the slope of the segment $(0,0)$ to $(x,y)$ is $\pm\sqrt{1+x}$, which means the derivative of the two branches, which is the limit as $x\to 0$ of this slope, is $\pm 1$.
