Differentiation 9 examples using the product and quotient rules
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1 1 Differentiation 9 eamples using the proct and quotient rules J A Rossiter Slides Anthon Rossiter
2 Introction The preious ideos hae gien a definition and concise deriation of differentiation from first principles. The aim no is to gie a numer of orked eamples for more challenging cases. Here the focus is on comining the proct and quotient rules, hile also utilising a tale of results for simple functions. ( u d Slides Anthon Rossiter ( u d
3 Tale of common results Slides Anthon Rossiter 3 a a 1 n n na a cos( sin( sin( cos( ( sec tan( c c ce e 1 log cosh( sinh( sinh( cosh( ( sin cos( ( cos ec ( cos cot( ec ( cos sin( sec(
4 4 NUMERICAL EXAMPLES KEY TECHNIQUES 1. Define all functions used in the proct and quotient rules, ith their associated deriaties, clearl.. Ensure the laout of the ork is uncluttered and unamiguous. This ill aoid man tpos. 3. Use knon results from a tale hereer possile. Slides Anthon Rossiter
5 Eample 1 5 Find the deriatie of: f log( ( ( ( u d u (log 3log p( t( ; p dt t dp Here is a proct of to functions, so e need the proct rule to differentiate this. p( ; t( (log 3log ; dp ; Slides Anthon Rossiter dt 3 3 (log 3log
6 Eample 1 - continued Find the deriatie of: f log( ( ( (log 3log Sustitute into quotient formulae ( 5 4 1; d u d 6 Slides Anthon Rossiter
7 Eample 7 Find the deriatie of: h g ( 4e sec(3 ( dh u d sec(3 p( t( ; d p dt t dp Here ( is a proct of to functions, so e need the proct rule to differentiate this. p( dp Slides Anthon Rossiter ; ; t( dt sec(3; 3sin(3 cos (3 Straight from the tale of knon results.
8 Eample - continued Find the deriatie of: h d u 4e u dh g( sec(3 ( 4e ; 3sin(3 cos (3 8e sec(3; ; From preious page. Straight from the tale of knon results. Sustitute into quotient formulae d 8 Slides Anthon Rossiter
9 Eample 3 Find the deriatie of: 6sin(0.5cos(0.5log(3 h g( e tan(0. ( dh u 9 d Here is a proct of three functions and ( is a proct of to functions, so e need the proct rule for oth. Hoeer, using tales of knon results, students ill see a possile doule angle formulae in the numerator hich ill simplif the oerall function. 3sin( log(3 h g( e tan(0. Slides Anthon Rossiter ( Net, use proct rule to find deriaties of and (.
10 Eample 3 3sin( log(3 h g( e tan(0. ( Find deriaties of and ( using the proct rule. 10 u 3sin( log(3 p( t( ; dt dp p( 3sin( ; p t dp cos( ; e tan(0. q( r( ; d q dr Slides Anthon Rossiter r dq q( dq e e ; ; t( dt r( dr log(3; 1 tan(0.; 0.sec Straight from the tale of knon results. (0.
11 Eample 3 3sin( log(3 h g( e tan(0. d ( 1 3sin( cos( log(3 e 0.sec (0. tan(0. Using results of preious page. e Finall, sustitute into quotient formulae. dh u d 11 Slides Anthon Rossiter
12 Summar 1 This ideo has demonstrated the differentiation of commonplace functions using a lookup tale in comination ith the proct and quotient rules. Vieers ill see that the most important points are: Keep clear definitions of functions used in the proct and quotient rules and their deriaties efore sustituting into the formulae. Use a lookup tale for common results. Don t orr if the algera gets mess, ut make sure the laout is clear and ell organised. Slides Anthon Rossiter
13 Anthon Rossiter Department of Automatic Control and Sstems Engineering Uniersit of Sheffield.shef.ac.uk/acse 016 Uniersit of Sheffield This ork is licensed under the Creatie Commons Attriution.0 UK: England & Wales Licence. To ie a cop of this licence, isit or send a letter to: Creatie Commons, 171 Second Street, Suite 300, San Francisco, California 94105, USA. It should e noted that some of the materials contained ithin this resource are suject to third part rights and an copright notices must remain ith these materials in the eent of reuse or repurposing. If there are third part images ithin the resource please do not remoe or alter an of the copright notices or esite details shon elo the image. (Please list details of the third part rights contained ithin this ork. If ou include our institutions logo on the coer please include reference to the fact that it is a trade mark and all copright in that image is resered.
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