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
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.
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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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Example 2 Find
the derivative of the following function using the definition of the
derivative.
Solution
This one is going to be a little messier as far as the
algebra goes. However, outside of that
it will work in exactly the same manner as the previous examples. First, we plug the function into the
definition of the derivative,
Note that we changed all the letters in the definition to
match up with the given function. Also
note that we wrote the fraction a much more compact manner to help us with
the work.
As with the first problem we can’t just plug in . So we will need to simplify things a
little. In this case we will need to
combine the two terms in the numerator into a single rational expression as
follows.
Before finishing this let’s note a couple of things. First, we didn’t multiply out the
denominator. Multiplying out the
denominator will just overly complicate things so let’s keep it simple. Next, as with the first example, after the
simplification we only have terms with h’s
in them left in the numerator and so we can now cancel an h out.
So, upon canceling the h
we can evaluate the limit and get the derivative.
The derivative is then,
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Example 3 Find
the derivative of the following function using the definition of the derivative.
Solution
First plug into the definition of the derivative as we’ve
done with the previous two examples.
In this problem we’re going to have to rationalize the
numerator. You do remember rationalization from an
Algebra class right? In an Algebra
class you probably only rationalized the denominator, but you can also
rationalize numerators. Remember that
in rationalizing the numerator (in this case) we multiply both the numerator
and denominator by the numerator except we change the sign between the two
terms. Here’s the rationalizing work
for this problem,
Again, after the simplification we have only h’s left in the numerator. So, cancel the h and evaluate the limit.
And so we get a derivative of,
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Let’s work one more example.
This one will be a little different, but it’s got a point that needs to
be made.
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Example 4 Determine
for
Solution
Since this problem is asking for the derivative at a
specific point we’ll go ahead and use that in our work. It will make our life easier and that’s
always a good thing.
So, plug into the definition and simplify.
We saw a situation like this back when we were looking at limits at infinity. As in that section we can’t just cancel the
h’s. We will have to look at the two one sided
limits and recall that
The two one-sided limits are different and so
doesn’t exist.
However, this is the limit that gives us the derivative that we’re
after.
If the limit doesn’t exist then the derivative doesn’t
exist either.
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In this example we have finally seen a function for which
the derivative doesn’t exist at a point.
This is a fact of life that we’ve got to be aware of. Derivatives will not always exist. Note as well that this doesn’t say anything
about whether or not the derivative exists anywhere else. In fact, the derivative of the absolute value
function exists at every point except the one we just looked at, .
The preceding discussion leads to the following definition.
Definition
The next theorem shows us a very nice relationship between
functions that are continuous and those that are differentiable.
Theorem
See the Proof
of Various Derivative Formulas section of the Extras chapter to see the
proof of this theorem.
Note that this theorem does not work in reverse. Consider and take a look at,
So, is continuous at but we’ve just shown above in Example 4
that is not differentiable at .
Alternate Notation
Next we need to
discuss some alternate notation for the derivative. The typical derivative notation is the
“prime” notation. However, there is
another notation that is used on occasion so let’s cover that.
Given a function all of the following are equivalent and
represent the derivative of with respect to x.
Because we also need to evaluate derivatives on occasion we
also need a notation for evaluating derivatives when using the fractional
notation. So if we want to evaluate the
derivative at x=a all of the
following are equivalent.
Note as well that on occasion we will drop the (x) part on the function to simplify the
notation somewhat. In these cases the
following are equivalent.
As a final note in this section we’ll acknowledge that
computing most derivatives directly from the definition is a fairly complex
(and sometimes painful) process filled with opportunities to make
mistakes. In a couple of sections we’ll
start developing formulas and/or properties that will help us to take the
derivative of many of the common functions so we won’t need to resort to the
definition of the derivative too often.
This does not mean however that it isn’t important to know
the definition of the derivative! It is
an important definition that we should always know and keep in the back of our
minds. It is just something that we’re
not going to be working with all that much.