MTH 251 – Differential Calculus

Chapter 3 – DifferentiationSection 3.2

The Derivative as a Function

Copyright © 2010 by Ron Wallace, all rights reserved.

Review – The Derivative at a Point• The derivative was defined as the limit of the

difference quotient. That is …

• If x0 + h = z, then an alternate definition would be …

• Note that the result of this limit is a number. That is, the derivative at a specific value of x.

0 00 0

( ) ( )' lim

h

f x h f xf x

h

Remember: x0 refers to a specific value of x.

0

00

0

( ) ( )' lim

z x

f z f xf x

z x

The Derivative as a Function

• If we do not specify a specific value of x (i.e. use x instead of x0) we get a function called the derivative of f(x).

• That is, the derivative of f(x) is the function …

0

( ) ( )'( ) limh

f x h f xf xh

( ) ( )'( ) limz x

f z f xf xz x

OR

f(x+h)

x x+hh

f(x)

f(x+h) – f(x)

Derivative Notation

• All of the following can be used to designate the function that is the derivative of y = f(x)

'f

'( )f x

'y

( )d f xdx

( )xD f x

dydx

dfdx

Reminder: The results of these will be a function.

Derivative at x = a Notation

• All of the following can be used to designate the derivative of y = f(x) at x = a

'( )f a ( )x a

d f xdx x a

dydx

( )x aD f x

a

dfdx

Reminder: The results of these will be a number.

Examples …

• Determine the following derivatives …

1ddx x

d xdx

nd xdx

IMPORTANT!Memorize these 3 results.

Examples …

• Determine the following derivatives …

7

1

x

ddx x

9

d xdx

5d xdx

Sketching the Graph of f ’(x) using the Graph of f(x)

• Where is the derivative (i.e. slope) zero?

• Where is the derivative (i.e. slope) positive? Large or small positive?

• Where is the derivative (i.e. slope) negative? Large or small negative?

• Where is the derivative (i.e. slope) constant? Function is a line segment. Derivative is a horizontal line segment.

Sketching the Graph of f ’(x) using the Graph of f(x)

• Example – Sketch the graph of the derivative of the following function.

Left & Right Derivatives at a Point• If in the definition of the derivative at a point,

you use just the left or right hand limit, the derivative at a point can be considered from just one side or the other.

• Right-Hand Derivative at x0

• Left-Hand Derivative at x0

• If these are equal, then …

0 00 0

' limh

f x h f xf x

h

0 00 0

' limh

f x h f xf x

h

00 0' ' 'f x f x f x

Left & Right Derivatives at a Point• Example:

24

' 2 ?

' 2 ?

f x x

f

f

Where does a derivative NOT exist?• Corner

left & right derivatives are different

22y x

Where does a derivative NOT exist?• Corner

• Cusp left & right derivatives

are approaching & –

21 52 4( 1)y x

Where does a derivative NOT exist?• Corner

• Cusp

• Vertical Tangent The derivative limit is

or –

31 1y x

Where does a derivative NOT exist?• Corner

• Cusp

• Vertical Tangent

• Discontinuity see the next theorem

3 2 65( 3)

x x xyx

Differentiability & Continuity

• If f ’(c) exists, then f(x) is continuous at x = c.

Proof …

Let x c h

Therefore: lim ( ) ( )x c

f x f c

0lim ( )h

f c h

0

lim ( ) [ ( ) ( )]h

f c f c h f c

0

[ ( ) ( )]lim ( )h

f c h f cf c hh

0 0 0

[ ( ) ( )]lim ( ) lim limh h h

f c h f cf c hh

( ) '( ) 0 ( )f c f c f c

as 0x c h

Differentiability & Continuity

• If f ’(c) exists, then f(x) is continuous at x = c.

Or … the contrapositive implies …

• If f(x) is NOT continuous at x = c, then f ’(c) does not exist.

• NOTES If the derivative does not exist, that does not

mean the function is not continuous. If the function is continuous, that does not mean

that the derivative exists. Example … the Absolute Value function.