Differentiation 2 first principles

Transcription

1 Differentiation first principles J A Rossiter Slides b Anthon Rossiter

2 Introduction The previous video introduces the concept of differentiation and the term derivative. Net we need to look at how differentiation is performed and the derivative computed. The focus here is on st principles, that is to show, briefl, how the main results are derived. Students who are happ 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. Slides b Anthon Rossiter

3 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! df Spoken as f derivative d d d f d. d d f Slides b Anthon Rossiter Spoken as d d of f. The action of differentiation.

4 What is differentiation? Differentiation is a process which finds the gradient of a curve, precisel, at an point along the curve. 4 /d = -.5 at this point. /d=.75 at this point /d =.75 at this point. Slides b Anthon Rossiter

5 First principles gradient estimation For a general curve, the gradient can be estimated using the formulae: gradient change change in in.5 Clearl not eact gradient. 5 gradient This is close, if difference between the -values is small. Slides b Anthon Rossiter

6 First principles gradient estimation For a general curve, the gradient can be estimated using the formulae: 5 Clearl not eact gradient gradient. 4 6 This is close, if difference between the -values is small. As difference gets smaller, the approimation becomes more accurate. Slides b Anthon Rossiter

7 First principles gradient estimation For a general curve, the gradient can be computed as a iting value: 5 [, ] d 4 Clearl, the smaller δ, the more accurate the gradient estimate. [, ] 7 Slides b Anthon Rossiter

8 We will not dwell on mathematical subtleties, but users need to assume the it eists and is well defined. For man curves, this it is not unique or well defined at some points and consequentl, at those points differentiation is not uniquel defined. Caviat d 8 Slides b Anthon Rossiter

9 9 EXAMPLES OF USING FIRST PRINCIPLES TO DERIVE DERIVATIVES OF SOME COMMON FUNCTIONS d Slides b Anthon Rossiter

10 Eample Simpl substitute into the formula from the previous page. d d d d Slides b Anthon Rossiter Visual inspection validates this answer is sensible.

11 Eample Simpl substitute into the formula. Slides b Anthon Rossiter d d d d Visual inspection validates this answer is sensible.

12 Eample Simpl substitute into the formula. d d n n n n n n n Ignore higher order terms in δ as these go to zero. d n n n n Slides b Anthon Rossiter

13 Eample 4 Simpl substitute into the formula. sina d d Slides b Anthon Rossiter sin a a sin a sin acos a cos asin a cos a ; sin a a; d sin a cos a a sin a cos a a a cos a d sin a

14 Eample 5 Simpl substitute into the formula. b e 4 d d e e b b b e b e e b b; d e b b be b Slides b Anthon Rossiter

15 Eample 6 log a This one is easiest handled b recognising the following relationship; this is obvious as it amounts to a simple swapping of the ais. d d or d d log a a e 5 d a e d a e Slides b Anthon Rossiter d a e a a

16 Table of some common results Slides b Anthon Rossiter 6 a d a n n na d a cos sin b b d b sin cos b b d b sec tan b b d b c c ce d e d log cosh sinh b b d b sinh cosh b b d b cot sin sin b b d b

17 Summar 7 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 ou. Later videos will graduall introduce known formulae and their application. Slides b Anthon Rossiter

18 Anthon Rossiter Department of Automatic Control and Sstems Engineering Universit of Sheffield 6 Universit of Sheffield This work is licensed under the Creative Commons Attribution. UK: England & Wales Licence. To view a cop of this licence, visit or send a letter to: Creative Commons, 7 Second Street, Suite, San Francisco, California 945, USA. It should be noted that some of the materials contained within this resource are subject to third part rights and an copright notices must remain with these materials in the event of reuse or repurposing. If there are third part images within the resource please do not remove or alter an of the copright notices or website details shown below the image. Please list details of the third part rights contained within this work. If ou include our institutions logo on the cover please include reference to the fact that it is a trade mark and all copright in that image is reserved.