Induction with derivatives

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December 30th 2012, 08:21 AM
zachd77

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Induction with derivatives

I've never been taught induction, but I want to attempt an induction proof with calculus.

I found these examples:

Find and prove a general formula for using the product rule for derivatives with n functions.

Prove by induction: Attachment 26412

How do I attempt both examples?
December 30th 2012, 09:07 AM
Plato

Re: Induction with derivatives

Quote:

Originally Posted by zachd77

I've never been taught induction, but I want to attempt an induction proof with calculus.
I found these examples:
Find and prove a general formula for using the product rule for derivatives with n functions.
Prove by induction: Attachment 26412

For induction this is the critical step:
$\frac{{d^{n + 1} }}{{dx^{n + 1} }}\left( {x^{n + 1} } \right) = \frac{{d^n }}{{dx^n }}\left[ {\frac{d}{{dx}}\left( {x^{n + 1} } \right)} \right]$ .
December 30th 2012, 10:20 AM
zachd77

Re: Induction with derivatives

I understand that step, but what would my base case be & how would I do the actual proof?
December 30th 2012, 10:51 AM
Plato

Re: Induction with derivatives

Quote:

Originally Posted by zachd77

I understand that step, but what would my base case be & how would I do the actual proof?

I will not do this question for you.

The base case is $\frac{{d^1 }}{{dx^1 }}\left( {x^1 } \right) = 1!$

Remember this is not a homework service.
December 30th 2012, 10:52 AM
zachd77

Re: Induction with derivatives

Sorry, this actually isn't a homework question. I was just curious. Sorry if I angered you slightly(Crying).
December 30th 2012, 12:33 PM
Siron

Re: Induction with derivatives

To answer the first question write down the derivative of the product of two function and three functions. Do you nottice a pattern? Thus, what can you conclude for $n$ functions?

Have you found a proof by induction for the second question now?
December 30th 2012, 03:40 PM
HallsofIvy

Re: Induction with derivatives

Quote:

Originally Posted by zachd77

Sorry, this actually isn't a homework question. I was just curious. Sorry if I angered you slightly(Crying).

Not anger but it is peculiar that you say "I want to attempt an induction proof with calculus" and then don't seem to make any attempt yourself.
December 30th 2012, 05:56 PM
hollywood

Re: Induction with derivatives

If you don't really understand how a proof by induction works, the example in the Wikipedia article on mathematical induction is pretty good.

- Hollywood
December 30th 2012, 08:34 PM
ibdutt

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Re: Induction with derivatives

Three steps are involved in Mathematical induction. Step 1: check that the statement is true for n = 1
Step 2: take that the statement is true for n = k
Step - 3: Prove that the statement is true for n = k+1 when it is true for n = k
If the first and last step are true i.e., the statement is true for n = 1 and statement being true for n=k implies that the statement is true for n = k+1 the we say the the statement is true for all values of n.
Attachment 26421
December 31st 2012, 05:32 AM
zachd77

Re: Induction with derivatives

Thanks.... I was never taught induction in my Precalc class, school schedule was tight so they unfortunately had to make cuts to the curriculum. I get the idea now, thanks for the help!

Also I should have reworded this post.. I didn't want to seem that this was a HW that I had to do. I'm just interested in the concepts that make calculus work. I'll just be careful in wording posts from now on (I'll attempt the problem first). Again, sorry if it seems that I'm coming for solutions for my HW. I'm not. I'm just a little curious and I want to go beyond the textbook.