Pagl!' 1 Ir ?"llll b~ Jilnicc L Ep5h:m Pagt" ? j i" 2111 J b~ Janice L Ep$ltln 31 Dtna11\'t
Derivatives (Section 3.1)
The derivative of a function f(x) at a number a, denoted by f' (a)
f'(a) = lim f(x) - f(a)
x ~a
X - a
::=
lim f(a
+ h) -
Interpretations Slope of the tangent line at x = a Instantaneous rate of change at x = a Velocity at x = a . cosx + 1
11m
b) X~31T X - 37r
3
+ 5x + 2
at a
=
""'~O
f:{I-t"h~ - r{\l J N,):.I"+5(1) \,-2­
~ (\') -::.. \ I Wl C\~?>
-
-e -'cS( W'r, ') \-2- -
\-..4tO
8
.
h~O
'
2.
31r ~ lOS311:=. \
'oj (\ 1(51= J2H'Y1 f(St-hj- £i;;.J­
')
\0-1D
Vl ::= \IVY) {'=> 4--h -\' -..rs=-L -::; 11m ~"1
~
_ -OJrn J. fAt:h -A' ~ -
h-=7D
f(x) h
It is differentiable on an open interval (a, b) if it is differentiable at
every number in the interval.
*¥ -2, .. r +-;z..
(I.r2..
vl~'tlh' 1-2-) -.JAVVJ
~ =..L
"'.., D 44-h \- 2
4
\
+ h) -
A functionfis differentiable at a if/,(a) exists.
~~O
n?O
-::::
Given a functionfix), we associate with it a new function/" called
the derivative off defined by the equation below
f'(x)' lim f(x
== \I N1 Lf it-~ n \-'e'l1'l-l,-\.,~ H'5' irS \-, -¥) ~~) 'V' \: :; \\m e>4- o\-) -\-'h'L.~ S
h~O
ct..==
/j
1
b) f(x)=~ at a = 5
,:1..) ! I (I'r= -.0.;.«1
yC'IJ -= 1-..3 evvJ.
=-7 ~Ly...):.. ~5
EXAMPLE 1 Find f'(a) for the given functions a) f (x) = x
EXAMPLE 2
Each limit represents the derivative of some functionfat some
number a. Statefand a in each case.
lim (2+h)3 -8
a) h~O
h
f(a)
h
h~O
3 I Dtnollvc
Iffis differentiable at a, thenfis continuous at a.
Pa~ 3 1 C 20 I J by Janice
L. EpstclIl 3 J Dervaliyc
EXAMPLE 3 At what values of x isfnot differentiable? I 1-~yv\
f(x)
(j
EXAMPLE 5
Find the derivative of the given function using the definition of
derivative. State the domain of the function and the domain of its
derivative.
n"UD-~~ ) ~o
')(-;::. -d.. ~ Old;­
\ _ )--rotA fMv.l\-~ umt I ~o L
-\J
Y. -;;. 0 ) -;( =="3 ~ Oo.T
~
~~
-
x
5
~ 5O.'f'I\Q~f\.2.l '(\. OY\
bbo/h~,~
)(::. - '3 \ ':,ou--r
..,
(b) f(x)
=
"or
tSl" ~
t!' - - - +-u~
~~~
­
~
'fC\
:II.. .+
-(I) (c)
r.-. le r
,
;
I
s\~ 0 ~-I-oIY'O
x+I x-I ----
f/
\
x
EXAMPLE 4
Match the graph of each function in (a) - (d) with the graph of its
derivation in I - IV. Explain
(c) [J:)
(a) ,.)
(b)
(d) ~ "I
=..)6 - x
(c) g(x) =_1
2
I
I
(a) f(x)
+
- 5
s11'.J'o
3.1 Dervaliw
Page 4 110 20 II by Jamcc L. Epstein (II) (Gt..)
(III)
(IV) (j?) r: &")
s)~'
I
cl~~'I1)
/,?.J
0 -­
t
-. -
-ff -
.
61ep ­
\
1
(d) G(t)=/t_I
Nlt.1 ~
o.)~Y\ ~ fl~) -~ ).{Co
~I("J~
~ rt-x+l")
- S:-l~] '" -hv'Y\ ~~ ­
h~o
. '9\
",,40
J am (~-1--h) - (fa.=>). =~ __
Y\ _
h... 0 1-'.0"" --it-h ,~) """"'7<:> h (
=
")
-=- "'''':;'0
~'rl'\ ~- o-')L_
\
~.. ~I'V-- -.-L­
.1 -,....,..-...-,"\,.
clnwuJ'\ C7t-- f 1("'-'" ~
b) f
X" b
r~"" L - ~l
h~O nL~-\--h-' )l.-ij
..
I(X)-=.0-<Y'I 1.
::: \ \V'Yj
\
r ')(__\
""'~D ~ L
':;:. Q.; W'l J.
" '...... 0
-\- -
v.
-\
J
-.,4+-\-'\ -,",) ()(-\J ,
,...
r~":.
1-\"2::-;')(
-t'- xl).+;()( -\,..\]
hL
yV - h -\
J
(01"",-,)(-.1-1 ")
=..Q;YV1l. r- zh
l- -!::­
\f\~O n LG:+n-O()l.-\)j-(t<--I)zA f'lY'VV'l I' ""
* of ::..
&{f"fY\{).J.. '(1
o~-r'
::;..
~.
..J.
ro