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§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
14
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
16
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,
18
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
19
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 $800.
20
Hw Problem56. Online Trading The profitability p (measured in quarterly net income) of the Charles Schwab Corporation increased as more of its customers traded online according to the formula
online) done tradesoffraction (
dollarsmillion 100300520 2
u
uuup
During that time, the fraction of online trades increased according to 1998) 1,January since months ( 020420 tt..tu
Use direct substitution to express the quarterly profits p as a function of time (do not simplify the expression) and then use the chain rule to estimate the rate of change of profitability at the beginning of September 1998. Be sure to specify the units.
21
Hints:
tdt
dt
dt
dt
dt
dP0204203000204200204201040 ......
Use direct substitution, we have
t..tu
uuup
020420 and
dollarsmillion 100300520Given 2
100020420300020420520 2 tttP ....
6020420820
0203000200204201040
t
t
...
....
Then,
ComputeMonth
million$?8tdt
dP
22
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)