2.2 Derivative
1. Definition of Derivative at a Point:
The derivative of the function fŸx at x a is defined as
U
fŸa h " fŸa provided the limit exists.
f Ÿa lim
hv0
h
If the limit exists, we say that f is differentiable at x a, otherwise, we say that f is not differentiable at
U
x a. An alternative form of f Ÿa :
U
fŸx " fŸa provided the limit exists.
f Ÿa lim
x"a
xva
Remarks:
U
a. From the definition of f Ÿa , we know the derivative of f at x a is the same as the slope of the
tangent line to the curve y fŸx at x a; and it is also the instantaneous rate of change of f at
x a.
U
b. When f is a position function of a moving object, f Ÿa gives the velocity of the object when x a.
Example Find the derivative of fŸx 2x 2 " 3x 4 at x 1.
U
fŸa h " fŸa :
a. Use the formula: f Ÿa lim hv0
h
2
i. fŸ1 2Ÿ1 " 3Ÿ1 4 3
2
2Ÿ1 h 2 " 3Ÿ1 h 4 " 3
fŸ1 h " fŸ1 2 4h 2h " 3 " 3h 4 " 3
ii.
h
h
h
2
hŸ2h
1 2h h 2h 1
h
h
fŸ1 h " fŸ1 lim hv0 Ÿ2h 1 1
iii. lim hv0
h
U
iv. f Ÿ1 1
U
fŸx " fŸa :
b. Use the formula: f Ÿa lim xva
x"a
2
i. fŸ1 2Ÿ1 " 3Ÿ1 4 3
2
2
fŸx " fŸ1 Ÿ2x " 1 Ÿx " 1 2x " 3x 4 " 3 2x " 3x 1 2x " 1
ii.
x"1
x"1
x"1
x"1
fŸx " fŸ1 lim xv1 Ÿ2x " 1 2Ÿ1 " 1 1
iii. lim xv1
x"1
U
iv. f Ÿ1 1
Example Find the derivative of fŸx x 1 at x "1.
2x " 1
U
fŸa h " fŸa :
a. Use the formula: f Ÿa lim hv0
h
i. fŸ"1 0 0
"3
"1 h 1 " 0
h
fŸ"1 h " fŸ"1 2Ÿ"1 h " 1
2h
"3
ii.
h
h
h
fŸ"1 h " fŸ"1 iii. lim hv0
lim hv0 1
"1
3
h
2h " 3
U
iv. f Ÿ"1 " 1
3
U
fŸx " fŸa :
b. Use the formula: f Ÿa lim xva
x"a
i. fŸ"1 0 0
"3
1
h
1
2h " 3
hŸ2h " 3 x1 "0
fŸx " fŸ"1 x1
1
2x
"1
ii.
x1
2x " 1
x1
Ÿx 1 Ÿ2x " 1 fŸx " fŸ"1 iii. lim xv"1
lim xv"1 1
"1
3
2x " 1
x1
Example Find the derivative of fŸx 3x " 2 at x 2.
U
fŸa h " fŸa :
a. Use the formula: f Ÿa lim hv0
h
i. fŸ2 3Ÿ2 " 2 4 2
ii.
fŸ2 h " fŸ2 h
3Ÿ2 h " 2 " 2
h
3Ÿ2 h " 2 " 2
h
3Ÿ2 h " 2 2
3Ÿ2 h " 2 2
3h
Ÿ3Ÿ2 h " 2 " 4
3
h 3Ÿ2 h " 2 2
3Ÿ2 h " 2 2
h 3Ÿ2 h " 2 2
fŸ2 h " fŸ2 3
3
iii. lim hv0
lim hv0
3 3
22
4
h
4 2
3Ÿ2 h " 2 2
U
iv. f Ÿ2 3
4
U
fŸx " fŸa :
b. Use the formula: f Ÿa lim xva
x"a
i. fŸ2 3Ÿ2 " 2 4 2
3x " 2 " 2
3x " 2 2
fŸx " fŸ2 3x " 2 " 2
ii.
x"2
x"2
Ÿx " 2 3x " 2 2
3Ÿx " 2 3x " 2 " 4
3
iii.
3x " 2 2
Ÿx " 2 3x " 2 2
Ÿx " 2 3x " 2 2
fŸx " fŸ2 3
3
iv. lim xv2
lim xv2
3
4
x"2
4 2
3x " 2 2
2. Definition of the Derivative Function:
U
The derivative of a function fŸx is the function f Ÿx defined as
U
fŸx h " fŸx provided the limit exists.
f Ÿx lim
hv0
h
The process of computing a derivative is called differentiation.
Example Without computing, find the derivative function fŸx for
a. fŸx c
b. fŸx mx b
U
a. f Ÿx 0 because the tangent line to the curve y c at any point x a is the line y a whose slope is
0.
U
b. f Ÿx m because the tangent line to the curve y mx b at any point x a is the line y mx b
whose slope is m.
U
Example Find f Ÿx if it exists where
a. fŸx 2x 2 " 3x 4
a.
2
b. fŸx x 1 for x p 1
2
2x " 1
c. fŸx 2x 1 for x u " 1
2
2Ÿx h 2 " 3Ÿx h 4 " Ÿ2x 2 " 3x 4 h
2
2
2
2x 4xh 2h " 3x " 3h 4 " 2x 3x " 4
h
2
hŸ4x 2h " 3 4xh
2h
"
3h
4x 2h " 3
h
h
U
fŸx h " fŸx lim Ÿ4x 2h " 3 4x " 3
f Ÿx lim
hv0
hv0
h
fŸx h " fŸx h
b.
Ÿx h 1
Ÿx h 1 Ÿ2x " 1 " Ÿx 1 Ÿ2x 2h " 1 " x1
2x " 1
2Ÿx h " 1
Ÿ2x 2h " 1 Ÿ2x " 1 h
h
2
2
2x
"
2x
"
2xh
"
2h
x
1
2xh
2x
"
x
"
h
"
1
"
2x
hŸ2x 2h " 1 Ÿ2x " 1 "h " 2h
"3h
"3
hŸ2x 2h " 1 Ÿ2x " 1 hŸ2x 2h " 1 Ÿ2x " 1 Ÿ2x 2h " 1 Ÿ2x " 1 U
fŸx h " fŸx "3
"3
f Ÿx lim
, for x p 1 .
lim
hv0
hv0 Ÿ2x 2h " 1 Ÿ2x " 1 2
h
Ÿ2x " 1 2
fŸx h " fŸx h
c.
2Ÿx h 1 " 2x 1
h
2Ÿx h 1 " 2x 1
fŸx h " fŸx h
h
2Ÿx h 1 2x 1
2Ÿx h 1 " Ÿ2x 1 h 2Ÿx h 1 2x 1
2h
2Ÿx h 1 2x 1
h
U
f Ÿx lim
hv0
U
2Ÿx h 1 2x 1
h
2x 2h 1 " 2x " 1
2Ÿx h 1 2x 1
2
2Ÿx h 1 2x 1
fŸx h " fŸx lim
hv0
h
2
2 2x 1
2
2Ÿx h 1 2x 1
1
, for x " 1 .
2
2x 1
3. Sketching the Graph of f Ÿx from the Given Graph of fŸx :
U
U
From the definition of f Ÿa , we know f Ÿa m tan at the point a, fŸa . So, for a given graph of f we
U
can sketch a rough graph of f Ÿx by estimate m tan at as many points as possible. The following are
U
U
examples of exact graphs of fŸx and f Ÿx . In class, we will show how to sketch rough graphs of f Ÿx from given fŸx .
3
6
15
4
10
2
5
-3
-4
-3
-2
-1
0
-2
-1
0
1
x
2
3
-2
1
2x
3
4
-4
-5
-6
U
U
- y fŸx , ... y f Ÿx - y fŸx , ... y f Ÿx 6
8
4
6
4
2
2
-4
-2
0
-2
2x
4
-3
-2
-1
0
1
x 2
3
-2
-4
-4
-6
-8
-6
U
- y fŸx , ... y f Ÿx U
- y fŸx , ... y f Ÿx Example Page 173: 21-26, 27
4. Alternative Derivative Notations:
The following are all alternative for the derivative function: let y fŸx U
dy
df d fŸx f Ÿx y U dx
dx
dx
d is also called a differential operator.
dx
5. Nondifferentiable Points of a Function:
U
From the definition of f Ÿa , we know the function is not differentiable at x a if the limit
fŸx h " fŸx does not exist.
lim
hv0
h
U
Note that the following are several conditions with which f Ÿa does not exist:
a. f is not continuous at x a ( fŸa is not defined; lim hv0 fŸx DNE or lim hv0 fŸx p fŸa ;
fŸx h " fŸx b. lim hv0
.;
h
fŸx h " fŸx DNE;
c. lim hv0
h
4
U
On the other hand, f is continuous at x a if f Ÿa exists.
Example Page 173. 33-34.
33. f is not differentiable at x "3, x "1 and x 3 since fŸx is not continuous at these x.
fŸx h " fŸx fŸx h " fŸx . f is not
34. f is not differentiable at x "2 since lim hv0 "
p lim hv0 h
h
differentiable at x 3.
x2
Example Let fŸx if x u 0
. Show graphically and algebraically that f is continuous at
"x
if x 0
x 0 but fŸx is not differenitable at x 0.
3
Graphically, we see f is continuous at x 0 and
2
is not differentiable at x 0.
1
-2
0
-1
1x
2
Algebraically, since lim xv0 fŸx lim xv0 Ÿx 2 0 and lim xv0 " fŸx lim xv0 " Ÿx 2 0, f is continuous at
fŸ0 h " fŸ0 fŸ0 h " fŸ0 and lim hv0 .
x 0. Now check lim hv0 "
h
h
h2 h
if h 0
fŸh " 0
fŸ0 h " fŸ0 h
h
h
"h "1
if h 0
h
fŸ0 h " fŸ0 fŸ0 h " fŸ0 lim" Ÿ"1 "1, lim
lim Ÿh 0
h
h
hv0
hv0
hv0
hv0
fŸ0 h " fŸ0 fŸ0 h " fŸ0 p lim hv0 Hence, f is not differentiable at x 0 since lim hv0 "
h
h
lim"
5