MATH 31 LESSONS Chapter 2: Derivatives 4. The Chain Rule

Section 2.6: The Chain Rule Read Textbook pp

A. Composite Functions (Review) A composite function is defined as where g (x) is the “inside function” f is the “outside function”

Ex. 1If f (x) = x 2 - 3x + 7 and g(x) = 4 - x 2, then find: Try this example on your own first. Then, check out the solution.

f (x) = x 2 - 3x + 7 g(x) = 4 - x 2 g (x) is the inside function. Replace it with g (x) = 4 - x 2

f (x) = x 2 - 3x + 7 g(x) = 4 - x 2 Wherever you see x in the f function, replace it with 4 - x 2

f (x) = x 2 - 3x + 7 g(x) = 4 - x 2

B. The Chain Rule For the composite function

First, take the derivative of the outside function (and leave the inside function the same)...

... then, take the derivative of the inside function

The chain rule can also be expressed in Leibnitz notation:

This is easy to remember, because if we treat these as true fractions, the du’s would cancel and you would be left with dy / dx. But of course, you would never do this.

The Chain Rule Applied to Power Functions The most common application of the chain rule in this unit is when the outside function is a power. e.g. y = [ f (x) ] n

First, take the derivative of the outside power function (and leave the inside function the same) then, take the derivative of the inside function

or

Ex. 2Differentiate using the chain rule. No need to simplify. Try this example on your own first. Then, check out the solution.

Method 1: Leibnitz Let u = x 2 - 3x Assign u as the “inside function”

Let u = x 2 - 3x Then, y = u 4 When you replace the inside function with u, you are left with just the outside function

u = x 2 - 3x y = u 4 This is the Leibnitz formula for the chain rule. Remember, to ensure it is in the proper form, you can “cancel” the du’s and you are left with dy / dx

u = x 2 - 3x y = u 4 Substitute y = u 4 and u = x 2 - 3x

u = x 2 - 3x y = u 4

u = x 2 - 3x y = u 4 Back substitute so that the answer is in terms of x

Method 2: “Outside, Inside” The “inside function” is x 2 - 3x

The “outside function” is the 4 th power

First, do the derivative of the outside function. Be certain to keep the inside function the same

Next, don’t forget to do the derivative of the inside function

Since this method is much faster, we will use this method exclusively from now on.

Ex. 3Differentiate using the chain rule. No need to simplify. Try this example on your own first. Then, check out the solution.

Bring all the x’s to the top.

First, do the derivative of the outside function. Be certain to keep the inside function the same

Next, don’t forget to do the derivative of the inside function

Ex. 4Differentiate using the chain rule. No need to simplify. Try this example on your own first. Then, check out the solution.

Express in power notation.

First, do the derivative of the outside function. Be certain to keep the inside function the same

Next, don’t forget to do the derivative of the inside function

Ex. 5Differentiate using the chain rule. No need to simplify. Try this example on your own first. Then, check out the solution.

First, do the derivative of the outside function. Be certain to keep the inside function the same

Next, don’t forget to do the derivative of the inside function

Apply the derivative to each part of the inside function. You will be required to do the chain rule again.

Derivative of “outside function” (leave inside same) Don’t forget the derivative of the “inside function”

Ex. 6Find Try this example on your own first. Then, check out the solution.

You read this as: “Find the derivative of y, and then evaluate it at x = 3”

 First, find the derivative using the chain rule:

chain rule

 Next, evaluate the derivative at x = 3:

Ex. 7If g (3) = 6, g (3) = 5, f (5) = 2, and f (6) = 8, then evaluate: Try this example on your own first. Then, check out the solution.

 Expand the function first in terms of x : First, do the derivative of the outside function. Be certain to keep the inside function the same

Next, don’t forget to do the derivative of the inside function

 Now, evaluate the function: g (3) = 6 g (3) = 5 f (5) = 2 f (6) = 8

g (3) = 6 g (3) = 5 f (5) = 2 f (6) = 8

g (3) = 6 g (3) = 5 f (5) = 2 f (6) = 8

Ex. 8Differentiate, using more than one rule. Fully factor your answer. Try this example on your own first. Then, check out the solution.

Which rule do you use first?

Take the derivative of the first and leave the second + Leave the first and take the derivative of the second (u  v) = u v + u v Use the product rule first

Next, use the chain rule

Put in the same order.

Let A = x Use substitution to make the factoring easier.

But A = x After factoring, back substitute so that it is in terms of only x. Be certain to use brackets.

But A = x Simplify inside the bracket.

But A = x 2 + 6

Ex. 9Differentiate, using more than one rule. Fully simplify your answer. Try this example on your own first. Then, check out the solution.

Which rule do you use first?

Chain rule first First, do the derivative of the outside function. Be certain to keep the inside function the same

Don’t forget to do the derivative of the “inside function”.

Use the quotient rule Quotient Rule:

This is another possible answer.