Product & Quotient Proofs

likely_lass

just a quick q im fairly certain you can do them by the lm method just want to make sur, does anyone know for sure?

i mean like lny=lnu-lnv

HQvhs

Unless they state specifically to use first principles then yes. Unfortunately most times they ask those proofs they've stated first principles. It's a pity cos the natural log method is so easy!

timmywex

likely_lass wrote: »

just a quick q im fairly certain you can do them by the lm method just want to make sur, does anyone know for sure?

i mean like lny=lnu-lnv

You can indeed, very easy method imo

likely_lass

thanks just looked up the papers(probably what i should have done first) your right they say first principals so i guess ill learn the other stupid way

muscle--museum

Watch out for the proof of the differentiation rule by induction =D simple if you've seen it once but impossible otherwise...

alan4cult

No you can always use the ln method although it ain't first principles.
There was lengthy discussion about this and the result can be read here
http://maths.slss.ie/resources/Reply%20to%20Query.doc

HQvhs

alan4cult wrote: »

No you can always use the ln method although it ain't first principles.
There was lengthy discussion about this and the result can be read here
http://maths.slss.ie/resources/Reply%20to%20Query.doc

Ah you're right! I didn't read the whole thing (I don't have that kind of time on my hands!) But it seems that they won't ask specifically for those proofs from first principles until they amend the syllabus, thereby making it quite possible that they may just ask you to just "Prove the product rule". As long as they don't specifically say "by first principles", which apparently if that response still applies they can't (it was December '07 so they may have revised it), you can use the log method.

Prowetod

muscle--museum wrote: »

Watch out for the proof of the differentiation rule by induction =D simple if you've seen it once but impossible otherwise...

??? Never heard of it. Is it in the text books? Or where could it be seen?

Nihilist21

eoccork wrote: »

??? Never heard of it. Is it in the text books? Or where could it be seen?

It's the very last proof at the back of "New Concise Maths 4" if you have that book.

Fringe

eoccork wrote: »

??? Never heard of it. Is it in the text books? Or where could it be seen?

It should be since it's one of the examinable proofs. It's not that bad. It's basically just proving that x^n differentiates to nx^n-1. Use induction.

MathsManiac

It was asked in 2006, so you'll find a proof in the marking scheme - p25 here:
http://www.examinations.ie/archive/markingschemes/2006/LC003ALPO00EV.pdf

Another pretty interesting proof (really a derivation I suppose) of the quotient rule, if anybody's interested (where u and v are the functions to be differentiatied):

Using the product rule, (uv)' - u'v + uv', and (v^-1)' = (-v^-2)v'

(u/v)' = (u.v^-1)'
= u'.v^-1 + u(v^-1)'
= u'.v^-1 + u(-1)v^-2.v'
= (u'/v) - (uv'/v^2)
= (u'v - uv')/v^2

You couldn't use it for a first principles question obviously. But it's handy to know none the less.

(P.S. - Only applies if product rule is assumed to be true/or proven before you use this proof. Just my disclaimer)