differentiation 2 first principles - university of...

Differentiation 2first principles

J A Rossiter

1

Slides by Anthony Rossiter

Introduction

• The previous video introduces the concept of differentiation and the term derivative.

• Next we need to look at how differentiation is performed and the derivative computed.

• The focus here is on 1st principles, that is to show, briefly, how the main results are derived.

• Students who are happy to go straight to core results without understanding the origins can skip this resource and go straight to resources 6 to get into some computations.


2

Recap

• Differentiation means to find the gradient; in general this involves some mathematical operations.

• A derivative is the result of differentiation, that is a function defining the gradient of a curve.

• The notation of derivative uses the letter ‘d’ and is not a fraction!


3

dx

dfderivative

dx

dyxfy )(

Spoken as ‘d f d x’.

)( fdx

d Spoken as ‘d d x of f’.The action of differentiation.

What is differentiation?Differentiation is a process which finds the gradient of a curve, precisely, at any point along the curve.


4

-2 -1.5 -1 -0.5 0 0.5 1-4

-3

-2

-1

0

1

2

x-2 -1.5 -1 -0.5 0 0.5 1

-4

-3

-2

-1

0

1

2

x

dy/dx= 1.75 at this point.

12 23 xxxy

dy/dx = -0.25 at this point.

dy/dx = 3.75 at this point.

First principles – gradient estimation

For a general curve, the gradient can be estimated using the formulae:


5

0.4 0.5 0.6 0.7 0.8 0.9 1-1.5

-1

-0.5

0

0.5

1

1.5

2

x

xinchange

yinchangegradient

12

12 )()(

xx

xyxygradient

This is close, if difference between the x-values is small.

Clearly not exact gradient.

0 0.5 1 1.5 2 2.5 3-10

0

10

20

30

40

50

x

First principles – gradient estimation

For a general curve, the gradient can be estimated using the formulae:


6

12

12 )()(

xx

xyxygradient

This is close, if difference between the x-values is small.

Clearly not exact gradient.

As difference gets smaller, the approximation

becomes more accurate.

0 0.5 1 1.5 2 2.5 3-10

0

10

20

30

40

50

x

First principles – gradient estimationFor a general curve, the gradient can be computed as a limiting value:


7

xxx

xyxxy

dx

dyx

)(

)()(lim 0

Clearly, the smaller δx, the more accurate the

gradient estimate.

)](,[ xxyxx

x)](,[ xyx

CaviatWe will not dwell on mathematical subtleties, but users need to assume the limit exists and is well defined.

For many curves, this limit is not unique or well defined at some points and consequently, at those points differentiation is not uniquely defined.


8

xxx

xyxxy

dx

dyx

)(

)()(lim 0

EXAMPLES OF USING FIRST PRINCIPLES TO DERIVE DERIVATIVES OF SOME COMMON FUNCTIONS


9

xxx

xyxxy

dx

dyx

)(

)()(lim 0

Example 1

Simply substitute into the formula from the previous page.


10

2xy

xxx

xyxxy

dx

dyx

)(

)()(lim 0

xxx

xxx

dx

dyx

)(

)(lim

22

0

xx

xxx

dx

dyx 2

))(2(lim 0

-2 -1 0 1 2 30

1

2

3

4

5

6

7

8

9

xVisual inspection validates this answer

is sensible.

x

xxxxx

dx

dyx

222

0

))(2(lim

Example 2

Simply substitute into the formula.


11

3xy

xxx

xyxxy

dx

dyx

)(

)()(lim 0

x

xxxxxxx

dx

dy

xxx

xxx

dx

dy

x

x

33223

0

33

0

))()(33(lim

)(

)(lim

222

0 3))()(33(

lim xx

xxxxx

dx

dyx

Visual inspection validates

this answer is sensible.

Example 3



12

nxy

x

xxnxx

dx

dy

xxx

xxx

dx

dy

nnn

x

nn

x

)(lim

)(

)(lim

1

0

0

11

0

)(lim

nn

x nxx

xnx

dx

dy

Ignore higher order terms in δx as these

go to zero.

Example 4



13

)sin(axy

x

axxaaxxaax

dx

dy

x

axxaax

dx

dy

x

x

)sin()sin()cos()cos()sin(lim

)sin()sin(lim

0

0

;)sin(lim;1)cos(lim 00 xaxaxa xx

)cos()cos(

lim 0 axax

xaax

dx

dyx

x

axxaaxax

dx

dyx

)sin()cos(1)sin(lim 0

Example 5



14

bxey

x

ee

dx

dy

x

ee

dx

dy

xbbx

x

bxxxb

x

)1(lim

lim

0

)(

0

;)1(lim 0 xbe xb

x

bxbx

x bex

xbe

dx

dy

0lim

Example 6

This one is easiest handled by recognising the following relationship; this is obvious as it amounts to a simple swapping of the axis.


15

axy log

y

y

e

a

dx

dye

ady

dx

1

yea

xaxy1

)log(

xax

a

e

a

dx

dyy

1

11

dy

dx

dx

dyor

dydxdx

dy

Table of some common results


16

adx

dyaxy 1 nn nax

dx

dyaxy

)cos()sin( bxbdx

dybxy )sin()cos( bxb

dx

dybxy

)(sec)tan( 2 bxbdx

dybxy cxcx ce

dx

dyey

xdx

dyxy

1log

)cosh()sinh( bxbdx

dybxy )sinh()cosh( bxb

dx

dybxy

)cot()sin(

1

)sin(

1x

bxb

dx

dy

bxy

Summary

• This video has introduced differentiation using first principles derivations.

• The derivatives of a few common functions have been given.

• Readers can use the same procedures to find derivatives for other functions but in general it is more sensible to access a table of answers which have been derived for you.

• Later videos will gradually introduce known formulae and their application.


17

© 2016 University of Sheffield

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Anthony RossiterDepartment of Automatic Control and

Systems EngineeringUniversity of Sheffieldwww.shef.ac.uk/acse

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