Now we come to the "Granddaddy" of differentiation
rules. In Chapter #1 you looked at composite functions and how
to build them. In this section, you will look at the way they
are differentiated. This rule is so important that in most calculus
books whole chapters are devoted to its explanation and proof.
Not only is it important to understand as a rule of differentiation,
it is the key to understanding the Method of Substitution used
in anti-differentiation.
As a review, you can make two functions, f(x)
and g(x), the composite function (f-g)(x)
= f(g(x)). The function g(x)
in this example, kind of "lives" inside the function
f. To create this new function, you found out in
Example #1.10, that you need to substitute g(x)
into the function defined as f(g(x)). The new function
is the composite function. The rate of change, or derivative,
of this new function is the product of the rates of change of
the functions, g(x) and f(g(x)) .
The following example may help you visualize this:
If a car can travel 6 times as fast as a man can run, and a plane can travel 10 times as fast as the car, then the plane can travel 6 x 10, or 60 times as fast as a man can run. The two rates are multiplied:
Before you attack the chain rule, the first part of the following example is presented to refresh your mind on composite functions. You see, the chain rule is all about recognizing composite functions. What you need to learn is how to look at these complex functions and be able to pick out the inner function from the outer function. What you are about to do is to build a function first to help you see the inner function.
This example reviews how to make a composite function. Then knowing that, you will work backwards to solve for the derivative of the new, composite function, you just created.
Let's say you have two functions defined as f(x) and g(x). First you will turn them into a composite function f(g(x)). The derivative of this new function will then be found by using the Chain Rule.
As you did in Example #1.10 use Wildcard Variables to define the variable x. Then define both f and g as User Defined Functions. Making a composite function using g(x) and f(x) gives you f(g(x)), your new function. The mechanics are shown below (you may want look at Composite Function in Chapter 1 for a guide).
Now you have the function that you will obtain the derivative of by using the Chain Rule.
It is pretty easy to see that the inner function here is the 3x^2 + 10x. You can see that from the composite above where you had g(x) inside f(x). Now for the part that may be harder to see the first time you look at a composite function. You know what this one is because you built it from scratch.
The best way to write this is like the following:
or
Where....
u is the inner function!
To find the derivative, all you need do is take the derivative of each of these functions and multiply them together.
Remember, the independent variable in the first equation is u and in the second equation it is x. Therefore, you need to set up your Derivative Ops accordingly.
When you do the substitutions, your Notebook will look like the following:
Don't worry about the way LiveMath places the equation, the answer you got using paper and pencil is the same.
You may ask; "Why not just expand the original equation and then use the Power Rule?". You can if you want. But I challenge you to do it that way by hand, or, even using LiveMath. It will be quite cumbersome. In fact, taking the answer and expanding will take some time in LiveMath too.
You can also set up your Notebook combining the two derivative as shown below. But watch out, you do not want LiveMath to make the substitution into the u of the first Derivative Op. This will happen if you try to substitute both functions at once.
Not this way.
In the example above, the Chain Rule was used in a form sometimes called "The General Power Rule for Functions". You backed into the concept by creating the composite function first and then took the derivative of this new function. This was to demonstrate to you the importance of the inner function, and to help you recognize a composite function.
There is something you should keep in mind at this point in the
chapter. All of the discussion thus far has an important academic
purpose. LiveMath can automatically solve derivatives without
you even knowing which rule was used, as was just demonstrated
at the end of Example #3.10. The purpose in showing you how LiveMath
does this will play an important part in your understanding of
the calculus. Not only is it important for knowledge sake, it
will help you down the line, understand other more advanced calculus
concepts. So even though the computer can do all the work for
you, try not to forget the underlying concepts involved.
As you differentiate different functions by hand, you will use
one, or in many cases, several of these rules to solve problems.
The trick is to pick the right one, or combination of them, to
do the job.
Hopefully the next example will solidify the concept in your mind.
If you are having a hard time getting started, look at it this way. How would you solve this equation if you only had a calculator and were given a value for x.
Wouldn't you first cube this value before you substituted it into the sine function? Sure you would. You would enter the number into your calculator; then you would cube the number; then you would push the sine button to get the final answer.
In fact, a calculator would give you a number like below.
You have found the inner function and it is x^3.
And the outer function is:
where 
Just pretend that you are solving the problem given a value for x. What would you do with your calculator first...second...third. By going through this little exercise, you should be able to find the inner function, or, in some cases, functions.
Back to the problem, find their respective derivatives.
To find the answer, all you need do is multiply the derivatives of each of these two functions together.
For a more complicated example click here.
Example
3.11 The Chain Rule
