3.2 Differentiability

Yes

YesYes

No

No

All Reals

0,∞⎡

⎣⎢

⎞

⎠⎟

3,∞⎡

⎣⎢

⎞

⎠⎟

3.2

5

To be differentiable, a function must be continuous and smooth.

Derivatives will fail to exist at:

corner cusp

vertical tangent discontinuity

( )f x x= ( )2

3f x x=

( ) 3f x x= ( ) 1, 0

1, 0

xf x

x

− <⎧= ⎨

≥⎩

→

Most of the functions we study in calculus will be differentiable.

→

Derivatives on the TI-89:

You must be able to calculate derivatives with the calculator and without.

Today you will be using your calculator, but be sure to do them by hand when called for.

Remember that half the test is no calculator.

→

3y x=Example: Find at x = 2.dy

dx

d ( x ^ 3, x ) ENTER returns23x

This is the derivative symbol, which is .82nd

It is not a lower case letter “d”.

Use the up arrow key to highlight and press .23x ENTER

3 ^ 2 2x x = ENTER returns 12

or use: ( )^ 3, 2d x x x = ENTER

→

Warning:

The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable.

Examples: ( )1/ , 0d x x x = returns −∞

( )( ), 0d abs x x x = returns 1±

→

Graphing Derivatives

Graph: ( )ln ,y d x x= What does the graph look like?

This looks like:1

yx

=

Use your calculator to evaluate: ( )ln ,d x x1

x

The derivative of is only defined for , even though the calculator graphs negative values of x.

ln x 0x >

→

Two theorems:

If f has a derivative at x = a, then f is continuous at x = a.

Since a function must be continuous to have a derivative, if it has a derivative then it is continuous.

→

( ) 1

2f a′ =

( ) 3f b′ =

Intermediate Value Theorem for Derivatives

Between a and b, must take

on every value between and .

f ′1

23

If a and b are any two points in an interval on which f is

differentiable, then takes on every value between

and .

f ′ ( )f a′

( )f b′

3.3 Rules for Differentiation

If the derivative of a function is its slope, then for a constant function, the derivative must be zero.

( ) 0d

cdx

=example: 3y =

0y′=

The derivative of a constant is zero.

→

If we find derivatives with the difference quotient:

→

( )2 22

0limh

x h xdx

dx h→

+ −=

( )2 2 2

0

2limh

x xh h x

h→

+ + −= 2x=

( )3 33

0limh

x h xdx

dx h→

+ −=

( )3 2 2 3 3

0

3 3limh

x x h xh h x

h→

+ + + −= 23x=

(Pascal’s Triangle)

2

4dx

dx

( )4 3 2 2 3 4 4

0

4 6 4limh

x x h x h xh h x

h→

+ + + + −= 34x=

2 3

We observe a pattern: 2x 23x 34x 45x 56x …

( ) 1n ndx nx

dx−=

examples:

( ) 4f x x=

( ) 34f x x′ =

8y x=

78y x′=

power rule

→

We observe a pattern: 2x 23x 34x 45x 56x …

( )d ducu c

dx dx=

examples:

1n ndcx cnx

dx−=

constant multiple rule:

5 4 47 7 5 35d

x x xdx

= ⋅ =

→

When we used the difference quotient, we observed that since the limit had no effect on a constant coefficient, that the constant could be factored to the outside.

(Each term is treated separately)

( )d ducu c

dx dx=

constant multiple rule:

sum and difference rules:

( )d du dvu v

dx dx dx+ = + ( )d du dv

u vdx dx dx

− = −

4 12y x x= +34 12y x′= +

4 22 2y x x= − +

34 4dy

x xdx

= −→

Example:Find the horizontal tangents of: 4 22 2y x x= − +

34 4dy

x xdx

= −

Horizontal tangents occur when slope = zero.34 4 0x x− =

3 0x x− =

( )2 1 0x x − =

( )( )1 1 0x x x+ − =

0, 1, 1x = −

Plugging the x values into the original equation, we get:

2, 1, 1y y y= = =

(The function is even, so we only get two horizontal tangents.)

→

-2

-1

0

1

2

3

4

-2 -1 1 2

-2

-1

0

1

2

3

4

-2 -1 1 2

4 22 2y x x= − +

-2

-1

0

1

2

3

4

-2 -1 1 2

4 22 2y x x= − +

2y =

-2

-1

0

1

2

3

4

-2 -1 1 2

4 22 2y x x= − +

2y =

1y =

-2

-1

0

1

2

3

4

-2 -1 1 2

4 22 2y x x= − +

-2

-1

0

1

2

3

4

-2 -1 1 2

4 22 2y x x= − +

First derivative (slope) is zero at:

0, 1, 1x = −

34 4dy

x xdx

= −

→

product rule:

( )d dv duuv u v

dx dx dx= + Notice that this is not just the

product of two derivatives.

This is sometimes memorized as: ( ) d uv u dv v du= +

( )( )2 33 2 5d

x x xdx⎡ ⎤+ +⎣ ⎦

( )5 3 32 5 6 15d

x x x xdx

+ + +

( )5 32 11 15d

x x xdx

+ +

4 210 33 15x x+ +

=( )2 3x + ( )26 5x + ( )32 5x x+ + ( )2x

4 2 2 4 26 5 18 15 4 10x x x x x+ + + + +

4 210 33 15x x+ +→

quotient rule:

2

du dvv ud u dx dx

dx v v

−⎛ ⎞=⎜ ⎟⎝ ⎠

or 2

u v du u dvd

v v

−⎛ ⎞=⎜ ⎟⎝ ⎠

3

2

2 5

3

d x x

dx x

++

( )( ) ( )( )( )

2 2 3

22

3 6 5 2 5 2

3

x x x x x

x

+ + − +=

+

→

Higher Order Derivatives:

dyy

dx′= is the first derivative of y with respect to x.

2

2

dy d dy d yy

dx dx dx dx

′′′= = =

is the second derivative.

(y double prime)

dyy

dx

′′′′′= is the third derivative.

( )4 dy y

dx′′′= is the fourth derivative.

We will learn later what these higher order derivatives are used for.