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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 we
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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