# PowerPoint Presentation by Cl3Kz1

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```									2.2 The derivative as a function
In Section 2.1 we considered the derivative at a fixed number a.
Now let number a vary. If we replace number a by a variable x,
then the derivative can be interpreted as a function of x :
f ( x  h)  f ( x )
f ( x)  lim
h 0            h

Alternative notations for the derivative:
dy df   d
f ( x)  y          f ( x)  Df ( x)
dx dx dx
D and d / dx are called differentiation operators.
dy / dx should not be regarded as a ratio.
4

3

2

1
y  f  x

0    1     2    3      4   5     6   7   8   9
3
The derivative
is the slope of   2
the original
1
function.
0    1     2    3      4   5     6   7   8   9

-1
y  f  x
-2
6
5
4
3
y  x 3  2

           
2

 x  h
2
1
3 x 3     2
-3 -2 -1 0
-1
1
x
2   3
y  lim
-2                       h 0                   h
-3
6
5
x  2 xh  h  x
2                 2           2
4
3
y  lim
h 0         h
2
1
0
y  lim 2 x  h
-3 -2 -1 0    1 2 3
-1     x
-2                                     h0
-3

y  2 x
-4
-5
-6
Differentiable functions

•A function f is differentiable at a if f ′(a) exists.
It is differentiable on an open interval (a,b) [ or
(a,) or (- , a) or (- , ) ] if it is differentiable
at every number in the interval.

•Theorem: If f is differentiable at a, then f is
continuous at a.

• Note: The converse is false: there are functions
that are continuous but not differentiable.
Example: f(x) = | x |
To be differentiable, a function must be continuous and
smooth.

Derivatives will fail to exist at:

f  x  x
2
f  x  x   3

corner                    cusp

1, x  0
f  x  
f  x  3 x                                                  1, x  0
vertical tangent          discontinuity

Higher Order Derivatives:
dy
y              is the first derivative of y with respect to x.
dx

dy d dy d 2 y                is the second derivative.
y          2
dx dx dx dx                     (y double prime)

dy
y                 is the third derivative.    We will learn
dx                                         later what these
higher order
d                                    derivatives are
 4
y              y is the fourth derivative.    used for.
dx

```
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