D
efinition of a
D
erivative
In Calculus, we talk about the rate at which a function is changing at a point.
Essentially, we are asked to find the slope of the line that is tangent to a curve at a
single
point. If you think about this for a moment, something should strike you as wrong.
One cannot find a single line with only one point. There are infinitely many lines that
pass through one point. Consider the line
ym
x
=
for any
m
, it passes through the origin.
We need two points to find the equation of a line. With two points, we can find the
slope between the two points. The equation for slope is
21
2
1
()
yy f
x f
x
y
xxx
xx
−
−
∆
==
∆−
−
.
Suppose that we wanted to find the slope of the tangent line at a point
x
. We said that
we need two points. So, let’s pick a second point just slightly to the right of
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 Fall '08
 staff
 Calculus, Derivative, Slope

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