1 §4.2 The Chain Rule (Pages 251~261) Composite Functions: The Chain Rule: Lectures 9 & 10.

1

§4.2 The Chain Rule (Pages 251~261)

• Composite Functions:• The Chain Rule:

dx

duun

dx

ud nn

1

:RuleChain dGeneralize

dx

duuf

dx

ufd.

:RuleChain

:notation aldifferentior

dx

du.

du

dy

dx

dy

5123 Ex. . xxh

Lectures 9 & 10

2

Define A Composite Function

And say that h is the composite of f and u. We read f(u(x)) as “f of u of x”

32

5

where,

32 see We

325 If

4

4

xxu

uuf

xufxfxh

xxh

3

Generalized Power Rule

1n nd duu n u

dx dx

Ex. 1 1 22 23 4 3 4d d

x x x xdx dx

1 221

3 4 6 42

x x x

2

3 2

3 4

x

x x

4

Generalized Power Rule Example7

2 1( )

3 5

xG x

x

6

2

3 5 2 2 1 32 1( ) 7

3 5 3 5

x xxG x

x x

66

2 8

91 2 12 1 137

3 5 3 5 3 5

xx

x x x

Ex. 3

5

The Chain Rule

( ) ( )d du

f u f udx dx

The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.

If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and

6

Chain Rule Examples

xxdx

d22

dx

duuu

dx

d22

23 3xxdx

d

dx

duuu

dx

d 23 3

2

11

xxdx

d

dx

du

uudx

d2

11

1 nn nxxdx

d dx

dunuu

dx

d nn 1

Original Rule Generalized Rule

Power Rule Generalized Power Rule

7

Why shall we accept the chain rule?

dx

duu

dx

duu

dx

duuuu

dx

du

dx

d22 .

dx

duuu

dx

d22

dx

duu

dx

duu

dx

duu

dx

duu

dx

duuu

udx

duu

dx

duuu

dx

du

dx

d

222

2

2223

32

2

.

. dx

duuu

dx

d 23 3

And so on for higher positive power of u. Similarly, we can use quotient rule to verify the chain rule for negative powers.

8

Chain Rule Example

Ex. 4 1003 xxdx

d

xxdx

dxx 3993100

13100 2993 xxx

131100or 299299 xxx

992 13100 x 993100 xx

9

Chain Rule Example

Ex. 5 13 xdx

d

13132

1 21 xdx

dx /

21132

3 / x

132

3or

x

2113 / xdx

d

10

Chain Rule Example

Ex. 6

xxdx

d2

1

xxdx

dxx

2221

12122

xxx

12 xx

dx

d

22 1

12or

xx

x 22

12

xx

x

11

Chain Rule Example

Ex. 7

1

12

2

x

x

dx

d

22

22

1

1212

x

xxxx

22

2222

1

1111

x

xxdxd

xxdxd

..

22 1

4

x

x

12

Harder Examples Using the Chain Rule

Ex. 8

xxdx

dxxxx

dx

d3131331 52452352 ...

? find ,31Given 3-52

dx

dyxxy .

452

53

31

9157or

xx

x

.

..

3152313 53452 .. . xxx

31152313 53452 x

dx

dxxx .. .

13

Harder Examples Using the Chain Rule

Ex. 9

21

222 or xxyxxy

? find ,Given 22 dx

dyxyx

y

x

2

12or

21

2

2

12

2

1212

2

1

xx

xxxx

xxdx

dxx

dx

dy

22

12

2

1Then

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Chain Rule in Differential Notation

If y is a differentiable function of u and u is a differentiable function of x, then

dy dy du

dx du dx

If y = u3, where u = 4x + 1, then

Quick Example

222 14121243 xuudx

du

du

dy

dx

dy

15

Chain Rule Example5 2 8 2, 7 3y u u x x Ex. 10

dy dy du

dx du dx

3 2 7556 6

2u x x

3 28 2 757 3 56 6

2x x x x

3 27 8 2140 15 7 3x x x x

Sub in for u

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Ex. 11 Marginal Product Precision Manufacturer is informed by a consultant that its annual profit is given by P = 200,000 + 4000q – 0.46q2 – 0.00001q3, where q is the number of surgical lasers it sells each year. The consultant also informs Precision that the number of surgical lasers it can manufacture each year depends on the number n of assembly-line workers it employs according to the equation q = 100n (i.e., each worker contributes 100 lasers per year). Use the chain rule to find the marginal product dP/dn.

Chain Rule Example

17

Solution: rule,chain theFrom

dn

dq

dq

dp

dn

dp

100 and 0000309204000 where 2 dn

dqqq

dq

dp..

2

2

003092000400

1000000309204000

Thus,

q.q,

q.q.

dn

dq

dq

dp

dn

dp

nn,

n.n,dn

dp

2

2

309200000400

100003010092000400 or,

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Ex. 12 Marginal Revenue Suppose a company’s weekly revenue R is given as a function of the unit price p, and p in turn is given as a function of weekly sales q by means of a demand equation). If

and

find the marginal revenue when sales are 1000 items per week.

Chain Rule Example

pricein increase $1per $401000

q

dp

dR

per week. sold item additionalper $201000

qdq

dp

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Solution:

have werule,chain By the

is revenue marginal The

dq

dp

dp

dR

dq

dR

dq

dR

sold item additionalper 80020401000 $qdq

dR

Units: revenue per item = revenue per $1 price increase price increase per additional item

Thus, if the price is lowered to increase the demand from 1000 to 1001 items per week, the weekly revenue will drop by approximately $800.

20

Hw Problem56. Online Trading The profitability p (measured in quarterly net income) of the Charles Schwab Corporation increased as more of its customers traded online according to the formula

online) done tradesoffraction (

dollarsmillion 100300520 2

u

uuup

During that time, the fraction of online trades increased according to 1998) 1,January since months ( 020420 tt..tu

Use direct substitution to express the quarterly profits p as a function of time (do not simplify the expression) and then use the chain rule to estimate the rate of change of profitability at the beginning of September 1998. Be sure to specify the units.

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Hints:

tdt

dt

dt

dt

dt

dP0204203000204200204201040 ......

Use direct substitution, we have

t..tu

uuup

020420 and

dollarsmillion 100300520Given 2

100020420300020420520 2 tttP ....

6020420820

0203000200204201040

t

t

...

....

Then,

ComputeMonth

million$?8tdt

dP

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65. Pollution An offshore oil well is leaking oil and creating a circular oil slick. If the radius of the slick is growing at a rate of 2 miles/hour, find the rate at which the area is increasing when the radius is 3 miles. (The area of a disc of radius r is A = r2.)

Chain Rule Example

Solution:? find , and

hr

miles2 :Given 2

dt

dArA

dt

dr

hr

miles4

hr

miles222 rr

dt

drr

dr

d

dt

dr

dr

dA

dt

dA

hr

miles12

hr

milesmiles 34

hr

miles4

2

miles 3 rdt

dAr

( HW Problem 66)