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Basic rules of differentiation
)()()()()()(
)()()()(
number. realany is where,)()(
constant.any is where,0)()(1
xvxuxfxvxuxfy
xucxfxucxfy
nxnxfxxfy
cxfcxfynn
Recall that
)( )( xfDdxdy
yxf x
all represent the derivative. All rules follow from the definition of the derivative.
Derivative of a constant
Examples
0)( cdxd
cxfy )(
00limlim)()(
lim)(000
hhh h
cch
xfhxfxf
02 dxdy
y
0)(3)( xfxf
Power rule1)( nn xnx
dxd
222
0
22
0
33223
0
33
00
3)33(lim)33(
lim
)33(lim
)(lim
)()(lim)(
xhxhxh
hxhxh
hxhxhhxx
hxhx
hxfhxf
xf
hh
h
hh
n is any real number
Not easy to prove in general. For 3)( xxfy
1)( nn xnxdxd
n is any real number
Examples
911010 1010)()( xxxfxxf
2612525 2525 xxxdxd
xxxxfxxxf
21
21
21
)()( 2/112/12/1
8/318/58/5
85
85 xxx
dxd
221 11
xxyx
xy
Derivative of a constant multiple of a function
)())(( xudxd
cxucdxd
)()()(
lim
)()(lim
)()(lim)(
)()(
0
00
xuch
xuhxuc
hxuchxuc
hxfhxf
xf
xucxf
h
hh
Examples
2/12/12/3
215
23
55 xxyxy
88777 4
77
41
41
41
41
xxx
dxd
xdxd
xdxd
Sum or difference rule
)]([)]([)]()([ xvdxd
xudxd
xvxudxd
Also applies to the sum and difference of more than two functions.
Examples
xxxxxdxd
xdxd
xxdxd
104)2(5455 332424
2/34/572/34/57 2541
4025
1041
85 xxxxxx
2/54/182/54/18 105105 xdxd
xdxd
xdxd
xxxdxd
Problem 1
The position of an object (in feet) at time t (in seconds) is
3. Find the time when the velocity is zero.
9279)( 23 tttts
1. Find the velocity.
27183)()( 2 tttstv
2. Find the velocity at t = 1 and t = 2.
ft/sec 327)2(18)2(3)2(
ft/sec 1227)1(18)1(3)1(2
2
s
s
033396327183)( 22 ttttttts
The velocity is zero at t = 3.
Sales of digital cameras
)61( 3.16)( 8766.0 tttN
where N(t) is measured in millions and t is measured in years, with t = 1 corresponding to 2001.
According to projections made in 2004, the worldwide shipment of digital point-and-shoot cameras are expected to grow in accordance with the rule:
a. How many digital cameras were sold in 2001?
3.16)1( N
16.3 million digital cameras were sold in 2001
)61( 3.16)( 8766.0 tttN
b. How fast were sales increasing in 2001?
1234.01234.0 2886.14)8766.0(3.16)( tttN
3.14)1( N
c. What were the projected sales in 2005?
8.6653.16)5( 8766.0 N
Increasing at the rate of 14.3 million cameras per year
66.8 million cameras
d. How fast were sales increasing in 2005?
7.11)5(2886.14)5( 1234.0 N
Increasing at the rate of 11.7 million cameras per year
