### The idea of the derivative of a function

First we can tell what the *idea* of a derivative is. But the
issue of *computing* derivatives is another thing entirely: a
person can understand the *idea* without being able to effectively
*compute*, and vice-versa.

Suppose that $f$ is a function of interest for some reason. We can
give $f$ some sort of ‘geometric life’ by thinking about *the set
of points $(x,y)$ so that*
$$f(x)=y$$
We would say that this describes a *curve* in the $(x,y)$-plane.
(And sometimes we think of $x$ as ‘moving’ from left to right,
imparting further intuitive or physical content to the story).

For some particular number $x_o$, let $y_o$ be the value
$f(x_o)$ obtained as output by plugging $x_o$ into $f$ as input. Then
the point $(x_o,y_o)$ is a point on our curve. The **tangent line**
to the curve **at** the point $(x_o,y_o)$ is a line passing through
$(x_o,y_o)$ and ‘flat against’ the curve. (As opposed to *crossing
it at some definite angle*).

The *idea* of the derivative $f'(x_o)$ is that it is *the slope of the tangent line* at $x_o$ to the curve. But this isn't
the way to *compute* these things...

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##### Calculus Refresher

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