Intuition|Difficulties | Rules | Examples

Differentiable Functions

Seminar „Hands-On Math for Computer Scientists“Saarbrücken, Feb. 2nd 2005

Daniel Beck, Sebastian Blohm

Intuition|Difficulties | Rules | Examples

Outline

• Solving exercises intuitively• Difficulties when solving the exercises• General rules for differentiability• Applying the rules

Intuition|Difficulties | Rules | Examples

The exercise

Determine which of the following functions are differentiable:– f(x)=x² – f(x)=1/x – f(x)=|x-1| – f(x)= √x with x≤4 and f(x)=x/4 + 1 with x>4

Intuition|Difficulties | Rules | Examples

When is a function f differentiable?

• A function f is differentiable at a point x0 if:

– It is continuous at – There exists a limit one limit of the difference

quotient:

• A function “f” is called differentiable (in I ) if it is diffenrentiable at every x0 ∈ I.

• When ist a function differentiable (over his domain I) ?

0

0

0

( ( ) ( ))lim

( )x x

f x f x

x x

Intuition|Difficulties | Rules | Examples

How do I check for differentiability at x0 ?

• If I have a plot of the function:

– Check if x0 has exactly one tangent.

• In the general case:– Check if f is continuous (in particular: no

jump)

– Check if

and both exist.

0 0 0 0

0 0

, ,0 0

( ( ) ( )) ( ( ) ( ))lim lim

( ) ( )

x x x x x x x x

f x f x f x f x

x x x x

Intuition|Difficulties | Rules | Examples

f(x)=x²

• Differentiable over R

Intuition|Difficulties | Rules | Examples

f(x) = 1/x

• Differentiable over \{0}

Intuition|Difficulties | Rules | Examples

f(x)=|x-1|

• Differentiable over \{1}

Intuition|Difficulties | Rules | Examples

f(x)= √x with x≤4 and f(x)=x/4 + 1 with x>4

Depending on the visualization, non-differentiable point might not be visible at all.

Intuition|Difficulties | Rules | Examples

Applying the definition

• Example : f(x)=x²

• So, the limes exists for all

• This was very easy!– But what about sin(x²) ?– This is clearly the wrong way!

0 0 0

2 20 0 0 0

00 0 0

( ( ) ( )) ( ) ( )( )lim lim lim 2

( ) ( ) ( )

x x x x x x

f x f x x x x x x xx

x x x x x x

0x

Intuition|Difficulties | Rules | Examples

Difficulties

Does anyone dare to calculate ?

PLUS: We cannot possibly calculate the limit of the difference quotient for all elements of the domain.

• How do I determine which points are crucial?• How do I prove that I did not miss a non-

differentiable point?

• Solution : apply some „cooking recipe”

0

2 20

0

(sin sin )lim

( )

x x

x x

x x

Intuition|Difficulties | Rules | Examples

General rules for checking differentiability

• Notation : – Predicates

• f is differentiable at• f is differentiable over I

– Functions:• Range of the function f on

interval I• : for

• : for

0

0

0

( ( ) ( ))lim

( )

x x

f x f x

x x

( ) :IRange g

:Id0

:xd

0x x

0x

0( )

xd f

0( )

xd f0x x

0x

0

0

0

( ( ) ( ))lim

( )

x x

f x f x

x x

Intuition|Difficulties | Rules | Examples

General rules for checking differentiability

Addition:

Substraction :

Multiplication:

Division:

( )I I

I

d f d g

d f g

( )I I

I

d f d g

d f g

( )I I

I

d f d g

d f g

( )

I I

I

d f d gf

dg

Intuition|Difficulties | Rules | Examples

General rules for checking differentiability

( )Id x x( )

( )I

I

d f

d f (cos )Id x(sin )Id x

( ) where

( )J

I

d fJ I

d f

Some special cases

Intuition|Difficulties | Rules | Examples

General rules (continued)

• Chain rule:

• Case splits:

( ) ( ) ( )

( )I J

I

Range g J d g d f

d g f

( ) ( ) ( ) ( ) lim( ( )) lim( ( ))

(if x k then else )

J L k k

x k x k

I

d g d f d f d f f x g x

d f g

Intuition|Difficulties | Rules | Examples

Example sin x²

2

2

( ) ( )

( ) ( sin( ))

(sin( ))

d x x d x xd x x d x x

d x

komp

sinmulIdId

Intuition|Difficulties | Rules | Examples

Example 1/x

\{0} \{0}

\{0}

(1) ( )

1

d d xDIV

dx

Intuition|Difficulties | Rules | Examples

Example √x with x≤4 and f(x)=x/4 + 1 with x>4

4 4

44

4 44 4

0 44

( ) ( 4)(1)( )

4 ( 1) ( ) lim( ) lim( 1) 4 4( ) ( 1)

4

4 14

x x

xx

x xx

x

d x dx dd x x

d d x xxd x d

xd if x then x else

Intuition|Difficulties | Rules | Examples

Example |x-1|

1 1

1 1 11 1

1 11 1

( ) (1)( 1) ( ) (1)

( 1) ( ( 1)) lim( 1) lim( ( 1))( ( 1)) ( 1)

(if x 1 then else )

1

x x

x x x

x xx x

I

d x dd x d x d

d x d x x xd x d x

d f gd x

( ( ) 0 ) I

I

d if f x then f else fABS

d f

FAILS!

Intuition|Difficulties | Rules | Examples

How to explain these rule in Active Math?

• With well explained sentances!• Example :

– f+g is differentiable if f an g are differentiable on I

– g(f) is differentiable if g is differentiable over the range of f and f is differentiable on I

– “If x<=k then f else g” is differentiable if f and g are differentiable, and if they have the same value and the same derivation then aproching k

– The calculations are a good visualization of the reasoning.

Intuition|Difficulties | Rules | Examples

Discussion

• Limits of the rule approach:– Counter example:

• is differentiable over !

– Do we want to stop when the first non-differentiable point is found? (Or do we want to modify the rules respectively.)

– One rule needed for each operation to be covered.

sin(| |)x