Lesson 14.pdf - 14 Derivatives Derivative of a function...

14. Derivatives
July 26, 2019
Derivative of a function Definition Let I ⊂ R be an interval, and let f : I → R be a
function.
1. The function f is differentiable at c ∈ I if the limit
lim
x → c
f ( x ) - f ( c )
x - c
(1)
exists.
2. The limit is called the derivative of f at c and is denoted by f
0
( c ).
3. If f is differentiable at each point x in a set S ⊆ I , then f is differentiable on S , and
the function f 0 : S → R is called the derivative of f .
[Examples and remark]
1. Another version of the limit (1):
lim
h → 0
f ( c + h ) - f ( c )
h
.
2. Let f ( x ) = k be constant.
For any c ∈ R ,
f ( c + h ) - f ( c )
h
=
k - k
h
=
0
h
= 0 .
Then f 0 ( c ) = lim
h → 0
f ( c + h ) - f ( c )
h
= 0.
116

3. Let f ( x ) = x
3
+ 2 x and c = 2.
f ( c + h ) - f ( c )
h
=
(2 + h ) 3 + 2(2 + h ) - 2 3 - 2 · 2
h
=
3 · 2 2 h + 3 · 2 h 2 + h 3 + 4 + 2 h - 2 · 2
h
= 14 + 6 h + h
2
-→ 14
as h → 0. Then f 0 (2) = 14.
4. Let f ( x ) = sin x , c = 0.
f ( c + h ) - f ( c )
h
=
sin h - sin 0
h
=
sin h
h
→ 1
as h → 0. Then f 0 (0) = 1.
5. Let f ( x ) = cos x , any c .
f ( c + h ) - f ( c )
h
=
cos ( c + h ) - cos c
h
=
- 2 sin
2 c + h
2
sin
h
2
h
→ - 2 sin c ·
1
2
= - sin c
as h → 0. Then f 0 ( c ) = - sin c .
Theorem 0.1 f is differentiable at c if and only if the sequence
f ( x
n
) - f ( c )
x
n
- c
converges whenever ( x
n
) is a sequence in I which converges to c , with x
n
6 = c for all n .
Furthermore, if f is differentiable, the limit is f 0 ( c ) .
[Example and remark]
1. Geometrically, for a function y = f ( x ), its derivative f 0 ( c ) equals to the slop of the
tangent line of its graph at the point ( c, f ( c )).
Therefore, for a given function, from its graph picture, if you find that there is no
unique tangent line at some point c , it indicates that f 0 ( c ) may not exist.
Then Theorem 0.1 may be used to show that the derivative f 0 ( c ) does not exist. (see
the example below)
117

2. Let f ( x ) = | x | defined on R .
-
6
@
@
@
@
@
You find that there are two tangent line of the graph of this function at x = 0, which
indicates that f 0 (0) may not exist. We now use Theorem 0.1 to prove it: Let x
n
=
1
n
and e x
n
= - 1
n
. We have x
n
→ 0 and e x
n
→ 0, and
f ( x
n
) - f (0)
x
n
- 0
=
1
n
- 0
1
n
- 0
= 1 ,
f ( e x
n
) - f (0)
e x
n
- 0
=
1
n
- 0
- 1
n
- 0
= - 1 .
By Theorem 0.1, if f 0 (0) exists, these two limits should be the same, but they are not.
Therefore, f 0 (0) doest not exists.
Differentiation and continuity For differentiation and continuity, they have the rela-
tionship
= ⇒
f
0
( c ) exists f is continuous at c
6⇐ =
For example, f ( x ) = | x | is continuous at x = 0, but f 0 (0) does not exist (see the example
above). Conversely, we have
Theorem 0.2 If f ( x ) is differentiable at x = c , then f ( x ) is continuous at x = c .