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,
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
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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.
Solution
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,
Notice that every term in the numerator that didn�t have an
h in it canceled out and we can now factor an h out of the
numerator which will cancel against the h in the denominator. After that
we can compute the limit.
So, the derivative is,
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