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I've learned in Euclidean Geometry that the tangent is a line which pass through only a point. For example, if someone ask me to find the tangent at this point $A$, I can easily say that the tangent can be drawn like this:

In higher dimensions it seems that the things are a little bit more complicated. For instance, if we have a curve (the helix)

$$\alpha(t):\mathbb R\to \mathbb R^3,\ \alpha(t)=(a\cos t,a\sin t,bt)$$

We can have a infinite lines which are tangent to a given point in the helix. However, we usually say that the tangent to a given point is the derivative $\alpha'(t)$:

What makes the derivative a so special tangent? there are a lot of tangents which passes through a given point, but we usually say that $\alpha'(t)$ is THE tangent, why?

Thanks in advance

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    $\begingroup$ Definition: The tangent line to a curve $C$ at a point $P$ is the line that a very tiny and nearsighted bug will think it is on, if it is sitting on $C$ at that point. Note that if it is sitting on $y=1-|x|$ at the point $x=0$, its bottom will tell it there is no tangent line. $\endgroup$ Dec 9, 2013 at 19:30
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    $\begingroup$ Be careful: your idea that "the tangent is a line which pass through only a point" is not quite right. The line $y=x$ passes through a single point of the line $y=-x$, and yet neither is tangent to the other. $\endgroup$ Dec 9, 2013 at 19:33

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First, you should be a little clearer: the "tangent at this point $A$" doesn't make sense until you say "the tangent to the circle at the point $A$", because there could be many other shapes that pass through $A$ and have a tangent there.

Next, the derivative isn't defined to be the tangent (or to "touch at only one point"); rather, it's defined by a limit $$ f'(a) = \lim_{h \rightarrow 0} \frac{f(a+h) - f(a)}{h} $$ whose value, if the limit exists, is the slope of a line that's tangent to the graph of $y = f(x)$ at the point $(a, f(a))$, in the sense that for nice enough functions $f$, if you look at the graph of $f$ near the point $(a, f(a))$, it's generally a smooth enough bit of curve that there's a tangent line to it in the sense that you're used to thinking about tangent lines.

The reason the derivative's defined that way is that it captures the notion of how the value $f(x)$ is changing as $x$ passes through $a$. If $f(a+h)$ happened to equal $f(a)$ for $h$ near $0$, then we could say that $f$ was "locally constant" and there's no change there, which would be reflected in the fact that the limit turns out to produce $f'(a) = 0$.

So on to 3D: we plot $\alpha(t)$, representing, for instance, the position of a bumblebee at time $t$. What should $\alpha'(a)$ represent? Some description of how $\alpha(t)$ is changing at time $t = a$. What's that mean? It means something like "in which direction, and how fast, is the bumblebee going?" If the bee is travelling in the spiral path shown in your diagram, then the arrow labelled $\alpha'(t)$ captures this notion, while some other line through the point $\alpha(t)$ -- a horizontal line, for instance -- does not. Since derivatives were developed to describe changes and rates of change, the former is the preferred line.

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When the given curve $\gamma$ is a circle, an ellipse, or even a helix, one can put forward a "geometrically intuitive" argument what the tangent at a point $P\in\gamma$ should be. But for a curve which is not generated by some elementary geometric process, but is given by an equation, a parametric representation, or as a complicated locus, it is by no means evident how to arrive "geometrically" at the tangent. The main reason for this is that there is no concept of "tangency", or the opposite: "transversality", without entering into differential calculus somehow.

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