Question

Question is attached..on calculus (derivatives) and induction for higher order derivatives

Answer 1

Im mainly lost on the part where it says to recall a fact about factorials and how its applied in the induction part (I know math induction just dont see the connection with the factorial)

Answer 2

It looks to me like this identity thing is quite attached to the binomial theorem / pascal's triangle sort of ideas.

If you look at a and b and compare to pascal's triangle: http://en.wikipedia.org/wiki/Pascal%27s_triangle

Which is related to the binomial theorem: http://en.wikipedia.org/wiki/Binomial_theorem

And the binomial theorem requires you to compute this binomial coefficient thing, which is where the factorial comes from: http://en.wikipedia.org/wiki/Binomial_coefficient

Answer 3

That identity they've given you at the end is probably something that helps you move from the assumed true case of k, to k+1 (the induction step), and helps you neaten up those factorial things, for doing that induction step.

Answer 4

actually, a first step might be to write the formula of part c using a form like the binomial theorem, and then see how parts a and b are special cases of that general form. Then you have two initial cases true (a, b) for induction, and you can assume true for all n, and then use the identity given (which should fit into the binomial type of form) to prove n+1.

Answer 5

I was able to find this..

Answer 6

sorry here..

Answer 7

it's a hard one.

Answer 8

If you're really stuck, then this page lays out the whole proof: http://www.proofwiki.org/wiki/Leibniz%27s_Rule/One_Variable

if you're tempted to get some hints or the whole thing or something...

Answer 9

thanks hehe

I found this one very useful

http://physicspages.com/2011/03/22/generalized-product-rule-leibnizs-formula/

but Im stuck where it says In the first term, we can shift the summation index by replacing k by to get k-1

Answer 10

yeah same thing in the one you gave..where it says 'For the first summation, we separate the case k=n and then shift the indices up by 1.'

Answer 11

At the bit where it says "for the first summation, we separate the case k=n ..." ?

They removed the term where k=n from the first summation, so they had something like:



Then they shifted the index, by doing something like:


Where a_k is just the term inside the summation, indexed by k. If you expand out that summation you'll see that it works fine. On that page they just did it all in one jump which is a bit confusing.