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The Definition of the Derivative
In the first
section
of the last chapter we saw that the computation of the slope of a
tangent line, the instantaneous rate of change of a function, and the instantaneous
velocity of an object at
all required us to compute the following limit.
We also saw that with a small change of notation this limit could also be written as,
(1)
This is such an important limit and it arises in so many places that we give it a name.
We call it a
derivative
. Here is the official definition of the derivative.
Definition
The
derivative of
with respect to
x
is the function
and is
defined as,
(2)
Note that we replaced all the
a
’s in
(1)
with
x
’s to acknowledge the fact that the
derivative is really a function as well. We often “read”
as “
f
prime
of
x
”.
Let’s compute a couple of derivatives using the definition.
Example 1
Find the derivative of the following function using the definition of the derivative.

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So, all we really need to do is to plug this function into the definition of the derivative,
(1)
, and do
some algebra. While, admittedly, the algebra will get somewhat unpleasant at times, but it’s just
algebra so don’t get excited about the fact that we’re now computing derivatives.
First plug the function into the definition of the derivative.
Be careful and make sure that you properly deal with parenthesis when doing the subtracting.
Now, we know from the previous chapter that we can’t just plug in
since this
will give us a division by zero error. So we are going to have to do some work. In this case that
means multiplying everything out and distributing the minus sign through on the second term.
Doing this gives,

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