We want to find the slope of the tangent line to a graph at the point

Let `P` = (`x`,`y`) and
`Q` := (`a`,`b`). Let

Then the slope of the line

Now, we
chose an arbitrary interval to be Delta-`x`. How does the
size of Delta-`x` affect our estimate of the slope of the
tangent line? The smaller Delta-`x` is, the more accurate
this approximation is. There is a wonderful animation of this by Douglas
Arnold. Look at it here.
You can see on the left of the animation how Delta-`x`
decreases, causing the secant line the approach the tangent, where it
zooms in on the right. Another animation of this (also from Douglas
Arnold) is here.

What we want to do is decrease the size of Delta-`x` as
much as possible. We do this by taking the limit as
Delta-`x` approaches zero. In the limit, assuming the limit
exists, we will find the exact slope of the tangent line to the curve at
the given point. This value is the derivative;

There are a few different, but equivalent, versions of this definition. In more general considerations,

This leads to three commonly used ways of expressing the definition of the derivative:

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watko@mit.edu Last modified October 14, 1999.