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We’ve taken a lot of derivatives over the course of the last
few sections. However, if you look back
they have all been functions similar to the following kinds of functions.
These are all fairly simple functions in that wherever the
variable appears it is by itself. What
about functions like the following,
None of our rules will work on these functions and yet some
of these functions are closer to the derivatives that we’re liable to run into
than the functions in the first set.
Let’s take the first one for example. Back in the section on the definition of the
derivative we actually used the definition to compute this derivative. In that section we found that,
If we were to just use the power rule on this we would get,
which is not the derivative that we computed using the
definition. It is close, but it’s not
the same. So, the power rule alone
simply won’t work to get the derivative here.
Let’s keep looking at this function and note that if we
define,
then we can write the function as a composition.
and it turns out that it’s actually fairly simple to
differentiate a function composition using the Chain Rule. There are two
forms of the chain rule. Here they are.
Chain Rule
Each of these forms have their uses, however we will work
mostly with the first form in this class.
To see the proof of the Chain Rule see the Proof of Various Derivative
Formulas section of the Extras chapter.
Now, let’s go back and use the Chain Rule on the function
that we used when we opened this section.
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Example 1 Use
the Chain Rule to differentiate  .
Solution
We’ve already identified the two functions that we needed
for the composition, but let’s write them back down anyway and take their
derivatives.
So, using the chain rule we get,
And this is what we got using the definition of the
derivative.
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In general we don’t really do all the composition stuff in
using the Chain Rule. That can get a
little complicated and in fact obscures the fact that there is a quick and easy
way of remembering the chain rule that doesn’t require us to think in terms of
function composition.
Let’s take the function from the previous example and
rewrite it slightly.
This function has an “inside function” and an “outside
function”. The outside function is the
square root or the exponent of depending on how you want to think of it and
the inside function is the stuff that we’re taking the square root of or
raising to the ,
again depending on how you want to look at it.
The derivative is then,
In general this is how we think of the chain rule. We identify the “inside function” and the
“outside function”. We then differentiate the outside function leaving the inside function alone and
multiply all of this by the derivative of the inside function. In its general form this is,
We can always identify the “outside function” in the
examples below by asking ourselves how we would evaluate the function. For instance in the R(z) case if we were to ask ourselves what R(2) is we would first evaluate the stuff under the radical and
then finally take the square root of this result. The square root is the last operation that we
perform in the evaluation and this is also the outside function. The outside function will always be the last
operation you would perform if you were going to evaluate the function.
Let’s take a look at some examples of the Chain Rule.
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Example 2 Differentiate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
Solution
(a)
It looks like the outside function is the sine and the
inside function is 3x2+x. The derivative is then.
Or with a little rewriting,
[Return to Problems]
(b)
In this case the outside function is the exponent of 50
and the inside function is all the stuff on the inside of the
parenthesis. The derivative is then.
[Return to Problems]
(c)
Identifying the outside function in the previous two was
fairly simple since it really was the “outside” function in some sense. In this case we need to be a little
careful. Recall that the outside function
is the last operation that we would perform in an evaluation. In this case if we were to evaluate this
function the last operation would be the exponential. Therefore the outside function is the
exponential function and the inside function is its exponent.
Here’s the derivative.
Remember, we leave the inside function alone when we
differentiate the outside function.
So, the derivative of the exponential function (with the inside left
alone) is just the original function.
[Return to Problems]
(d)
Here the outside function is the natural logarithm and the
inside function is stuff on the inside of the logarithm.
Again remember to leave the inside function along when
differentiating the outside function.
So, upon differentiating the logarithm we end up not with 1/x but instead with 1/(inside
function).
[Return to Problems]
(e)
In this case the outside function is the secant and the
inside is the .
In this case the derivative of the outside function is . However, since we leave the inside function
alone we don’t get x’s in
both. Instead we get in both.
[Return to Problems]
(f)
There are two points to this problem. First, there are two terms and each will
require a different application of the chain rule. That will often be the case so don’t expect
just a single chain rule when doing these problems. Second, we need to be very careful in
choosing the outside and inside function for each term.
Recall that the first term can actually be written as,
So, in the first term the outside function is the exponent
of 4 and the inside function is the cosine.
In the second term it’s exactly the opposite. In the second term the outside function is
the cosine and the inside function is . Here’s the derivative for this function.
[Return to Problems]
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There are a couple of general formulas that we can get for
some special cases of the chain rule.
Let’s take a quick look at those.
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Example 3 Differentiate
each of the following.
(a)
(b)
(c)
Solution
(a) The outside
function is the exponent and the inside is g(x).
(b) The outside
function is the exponential function and the inside is g(x).
(c) The outside
function is the logarithm and the inside is g(x).
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The formulas in this example are really just special cases
of the Chain Rule but may be useful to remember in order to quickly do some of
these derivatives.
Now, let’s also not forget the other rules that we’ve got
for doing derivatives. For the most part
we’ll not be explicitly identifying the inside and outside functions for the
remainder of the problems in this section.
We will be assuming that you can see our choices based on the previous
examples and the work that we have shown.
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Example 4 Differentiate
each of the following.
(a) [Solution]
(b) [Solution]
Solution
(a)
This requires the product rule and each derivative in the
product rule will require a chain rule application as well.
In this part be careful with the inverse tangent. We know that,
When doing the chain rule with this we remember that we’ve
got to leave the inside function alone.
That means that where we have the in the derivative of we will need to have .
[Return to Problems]
(b)
In this case we will be using the chain rule in concert
with the quotient rule.
These tend to be a little messy. Notice that when we go to simplify that
we’ll be able to a fair amount of factoring in the numerator and this will
often greatly simplify the derivative.
After factoring we were able to cancel some of the terms
in the numerator against the denominator.
So even though the initial chain rule was fairly messy the final
answer is significantly simpler because of the factoring.
[Return to Problems]
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The point of this last example is to not forget the other
derivative rules that we’ve got. Most of
the examples in this section won’t involve the product rule or the quotient
rule to make the problems a little shorter.
However, in practice they will often be in the same problem.
Now, let’s take a look at some more complicated examples.
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Example 5 Differentiate
each of the following.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
Solution
We’re going to be a little more careful in these problems
than we were in the previous ones. The
reason will be quickly apparent.
(a)
In this case let’s first rewrite the function in a form
that will be a little easier to deal with.
Now, let’s start the derivative.
Notice that we didn’t actually do the derivative of the
inside function yet. This is to allow
us to notice that when we do differentiate the second term we will require
the chain rule again. Notice as well
that we will only need the chain rule on the exponential and not the first
term. In many functions we will be
using the chain rule more than once so don’t get excited about this when it
happens.
Let’s go ahead and finish this example out.
Be careful with the second application of the chain
rule. Only the exponential gets
multiplied by the “-9” since that’s the derivative of the inside function for
that term only. One of the more common
mistakes in these kinds of problems is to multiply the whole thing by the
“-9” and not just the second term.
[Return to Problems]
(b)
We’ll not put as many words into this example, but we’re
still going to be careful with this derivative so make sure you can follow
each of the steps here.
As with the first example the second term of the inside
function required the chain rule to differentiate it. Also note that again we need to be careful
when multiplying by the derivative of the inside function when doing the
chain rule on the second term.
[Return to Problems]
(c)
Let’s jump right into this one.
In this example both of the terms in the inside function
required a separate application of the chain rule.
[Return to Problems]
(d)
We’ll need to be a little careful with this one.
This problem required a total of 4 chain rules to
complete.
[Return to Problems]
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Sometimes these can get quite unpleasant and require many
applications of the chain rule.
Initially, in these cases it’s usually best to be careful as we did in
this previous set of examples and write out a couple of extra steps rather than
trying to do it all in one step in your head.
Once you get better at the chain rule you’ll find that you can do these
fairly quickly in your head.
Finally, before we move onto the next section there is one
more issue that we need to address. In
the Derivatives of Exponential and Logarithm
Functions section we claimed that,
but at the time we
didn’t have the knowledge to do
this. We now do. What we needed was the Chain Rule.
First, notice that using a property of logarithms we can
write a as,
This may seem kind of silly, but it is needed to compute the
derivative. Now, using this we can write
the function as,
Okay, now that we’ve gotten that taken care of all we need
to remember is that a is a constant
and so is also a constant. Now, differentiating the final version of
this function is a (hopefully) fairly simple Chain Rule problem.
Now, all we need to do is rewrite the first term back as to get,
So, not too bad if you can see the trick to rewrite a and with the Chain Rule.