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

MATH 31 LESSONS

Chapter 2:

Derivatives

4. The Chain Rule

Section 2.6: The Chain Rule

Read Textbook pp. 96 - 102

A. Composite Functions (Review)

A composite function is defined as

g (x) is the “inside function”

f is the “outside function”

xgfxgf

Ex. 1 If f (x) = x2 - 3x + 7 and g(x) = 4 - x2 ,

then find:

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

f (x) = x2 - 3x + 7

g(x) = 4 - x2

xgfxgf

g (x) is the inside function.

Replace it with g (x) = 4 - x2

f (x) = x2 - 3x + 7

g(x) = 4 - x2

xgfxgf

7434 222 xx

Wherever you see x in the f function, replace it with 4 - x2

f (x) = x2 - 3x + 7

g(x) = 4 - x2

xgfxgf

7434 222 xx

7312816 242 xxx

115 24 xx

B. The Chain Rule

For the composite function xgfxgf

xgxgfxgfdx

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

xgxgfxgfdx

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:

:andIf xguxgfy

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.

y = [ f (x) ] n

nxfy If

xfxfny n 1

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

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

nxfy If

xfxfny n 1

:andIf xguuy n

duuny n 1

Ex. 2 Differentiate using the chain rule.

No need to simplify.

42 3xxy

Method 1: Leibnitz

Let u = x2 - 3x

42 3xxy

Assign u as the “inside function”

Let u = x2 - 3x

Then, y = u4

42 3xxy

When you replace the inside function with u, you are left with just the outside function

u = x2 - 3x

y = u4

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 = x2 - 3x

y = u4

Substitute y = u4 and u = x2 - 3x

u = x2 - 3x

y = u4

324 3 xu

u = x2 - 3x

y = u4

324 3 xu

3234 2 xxx Back substitute so that the answer is in terms of x

Method 2: “Outside, Inside”

42 3xxy

The “inside function” is x2 - 3x

42 3xxy

The “inside function” is x2 - 3x

The “outside function” is the 4th power

42 3xxy

First, do the derivative of the outside function.

Be certain to keep the inside function the same

32 34 xxy

42 3xxy

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

dxxy 334 232

42 3xxy

dxxy 334 232

323432 xxx

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

Bring all the x’s to the top.

721 5268 xxxf

21721 5265268 xxdx

32721 251205268 xxxx

32721 1025268 xxxx

23 74 tttg

Express in power notation.

2323 7474 tttttg

23 742

2323 7474 tttttg

23 74742

2323 7474 tttttg

23 74742

tttt 1412742

9424 32 xxxy

8424 329 xxxy

9424 32 xxxy

4248424 32329 xxx

4248424 32329 xx

Apply the derivative to each part of the inside function.

You will be required to do the chain rule again.

4248424 32329 xxx

4248424 32329 xx

8424 3342329 xxdx

dxxxxx

Derivative of “outside function”

(leave inside same)

Don’t forget the derivative of the “inside function”

4248424 32329 xxx

4248424 32329 xx

8424 3342329 xxdx

dxxxxx

xxxxxxx 64342329 33248424

Ex. 6 Find

2and5if 23 xuuuy

You read this as:

“Find the derivative of y,

and then evaluate it at x = 3”

First, find the derivative using the chain rule:

2123 25 x

1103 2

12 xdx

chain rule

1103 2

12 xdx

1103 2

12 xuu

Next, evaluate the derivative at x = 3:

123,3At ux

11013 2

Ex. 7 If g (3) = 6 , g (3) = 5, f (5) = 2 , and f (6) = 8,

then evaluate:

Expand the function first in terms of x :

xgfxgf

Now, evaluate the function: g (3) = 6 g (3) = 5

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

xgxgfxgf

333 ggfgf

g (3) = 6 g (3) = 5

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

xgxgfxgf

333 ggfgf

g (3) = 6 g (3) = 5

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

xgxgfxgf

333 ggfgf

Ex. 8 Differentiate, using more than one rule.

Fully factor your answer.

523 6 xxy

Which rule do you use first?

523 6 xxy

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

523523 66 xxxxy

Use the product rule first

523 6 xxy

523523 66 xxxxy

66563 2423522 xdx

Next, use the chain rule

523 6 xxy

523523 66 xxxxy

66563 2423522 xdx

xxxxx 26563423522

424522 61063 xxxx

Put in the same order.

Let A = x2 + 6

xxxxx 26563423522

424522 61063 xxxx

4452 103 AxAx

242 103 xAAx Use substitution to make the factoring easier.

But A = x2 + 6

242 103 xAAx

22422 10636 xxxx

After factoring, back substitute so that it is in terms of only x.

Be certain to use brackets.

But A = x2 + 6

242 103 xAAx

22422 10636 xxxx

22422 101836 xxxx

Simplify inside the bracket.

But A = x2 + 6

242 103 xAAx

22422 10636 xxxx

22422 101836 xxxx

18136 2422 xxx

Ex. 9 Differentiate, using more than one rule.

Fully simplify your answer.

Which rule do you use first?

Chain rule first

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

xy Use the

quotient rule

Quotient Rule:

xxx This is another possible answer.

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

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