I am currently working on implicit differentiation, and I am stuck with interpreting what specifically "in exactly one other point" means. Does this literally mean showing the derivative, or is there something that I am completely missing? Any help would be greatly appreciated.
Here is the problem verbatim:
Show that at every point except for $(x,y) = (0,0), (\pm1,0)$, the tangent line to the curve $y^2 = x^3 - x$ meets the curve in exactly one other point.
Thank you very much to anyone who is willing to help me interpret this question.
They mean that for a general point $P$ on that curve other than those three points, the tangent line to the curve at $P$ intersects the curve in exactly one other point. So, you need to (a) find the slope of the curve at $P$, (b) construct the tangent line, and (c) show that the line intersects the curve at exactly two points, one of them $P$—except when $P$ is either $(0,0)$ or $(\pm1, 0)$.
It may be instructive to look at Diagram $2$ in the "group law" section of the Wikipedia plot summary for elliptic curves. This problem is in fact about an elliptic curve, and is probably motivated by that.