Intuition|Difficulties | Rules | Examples Differentiable Functions Seminar „Hands-On Math for Computer Scientists“ Saarbrücken, Feb. 2nd 2005 Daniel Beck, Sebastian Blohm
Dec 25, 2015
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)= √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 √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