# Induction with derivatives

• Dec 30th 2012, 08:21 AM
zachd77
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?
• Dec 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:
$\displaystyle \frac{{d^{n + 1} }}{{dx^{n + 1} }}\left( {x^{n + 1} } \right) = \frac{{d^n }}{{dx^n }}\left[ {\frac{d}{{dx}}\left( {x^{n + 1} } \right)} \right]$.
• Dec 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?
• Dec 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 $\displaystyle \frac{{d^1 }}{{dx^1 }}\left( {x^1 } \right) = 1!$

Remember this is not a homework service.
• Dec 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).
• Dec 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 $\displaystyle n$ functions?

Have you found a proof by induction for the second question now?
• Dec 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.
• Dec 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
• Dec 30th 2012, 08:34 PM
ibdutt
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
• Dec 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.