differentiation 8 examples using the quotient...

Differentiation 8examples using the quotient rule

J A Rossiter

1

Slides by Anthony Rossiter

Introduction

• The previous videos have given a definition and concise derivation of differentiation from first principles.

• The aim now is to give a number of examples.

• Here the focus is on the quotient rule in combination with a table of results for simple functions.


2

)(

)(

xv

xuy

2v

dx

dvu

dx

duv

dx

dy

Table of common results


3

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

NUMERICAL EXAMPLES


4

KEY TECHNIQUES

1. Define all functions used in the quotient rule, with their associated derivatives, clearly.

2. Ensure the layout of the work is uncluttered and unambiguous. This will avoid many typos.

3. Use known results from a table wherever possible.

Example 1

Find the derivative of:


5

)(

)(

)5cos(

2)(

4

xv

xu

x

xxxfy

dx

duxxu ;2 4

dx

dv

xv );5cos(

2v

dx

dvu

dx

duv

dx

dy

Define u,v and du/dx, dv/dx.

)5(cos

)5sin()2(5)18)(5cos(2

43

x

xxxxx

dx

dy

Example 2



6

)(

)()3tan()2sin(3)(

4.0 xv

xu

e

xxxfy

x

dx

duxxu );3tan()2sin(3

dx

dv

ev x ;4.0

2v

dx

dvu

dx

duv

dx

dy

Define u,v and du/dx, dv/dx.

x

xx

e

exxxxe

dx

dy8.0

4.024.0 4.0))3tan()2sin(3())3(sec3)2cos(6(

Example 3



7

)(

)(

)14(

)2log()(

5

3

xv

xu

xx

xxfy

xdx

duxu

3);log32(log

dx

dv

xxv ;145

2v

dx

dvu

dx

duv

dx

dy

SPECIAL CASES:INVERSE OF TRIGONOMETRIC FUNCTIONSINVERSE POLYNOMIALS


8

Example 4



9

)(

)(1)(

xv

xu

xxfy

n

0;1 dx

duu

1; nn nxdx

dvxv

2v

dx

dvu

dx

duv

dx

dy

For completeness


10

54

43

32

2

41

31

21

11

xxdx

d

xxdx

d

xxdx

d

xxdx

d

Example 5



11

)(

)(

)sin(

1)(

xv

xu

xxfy

0;1 dx

duu )cos();sin( x

dx

dvxv

2v

dx

dvu

dx

duv

dx

dy

)(sin

)cos()sin(02 x

xx

dx

dy

Example 6



12

)(

)(

)cos(

1)(

xv

xu

xxfy

0;1 dx

duu )sin();cos( x

dx

dvxv

2v

dx

dvu

dx

duv

dx

dy

)(cos

)sin()cos(02 x

xx

dx

dy

Example 7



13

)(

)(

)tan(

1)(

xv

xu

xxfy

0;1 dx

duu

)(cos

1)(sec);tan(

2

2

xx

dx

dvxv

2v

dx

dvu

dx

duv

dx

dy

)(tan

)(sec)tan(02

2

x

xx

dx

dy

Summary

• This video has demonstrated the differentiation of commonplace functions using a lookup table in combination with the quotient rule.

• Viewers will see that the steps are largely mechanical, albeit tedious at times.

• Keep clear definitions of u(x),v(x) and their derivatives before substituting into the formulae.

• In general it is advisable to have a lookup table to hand, even though in due course you are likely to remember many of the results.


14

© 2016 University of Sheffield

This work is licensed under the Creative Commons Attribution 2.0 UK: England & Wales Licence. To view a copy of this licence, visit http://creativecommons.org/licenses/by/2.0/uk/ or send a letter to: Creative Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA.

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Anthony RossiterDepartment of Automatic Control and

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

http://creativecommons.org/licenses/by/2.0/uk/

differentiation 8 examples using the quotient...

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