1. ## 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:

How do I attempt both examples?

2. ## Re: Induction with derivatives

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:

For induction this is the critical step:
$\displaystyle \frac{{d^{n + 1} }}{{dx^{n + 1} }}\left( {x^{n + 1} } \right) = \frac{{d^n }}{{dx^n }}\left[ {\frac{d}{{dx}}\left( {x^{n + 1} } \right)} \right]$.

3. ## Re: Induction with derivatives

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

4. ## Re: Induction with derivatives

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.

5. ## Re: Induction with derivatives

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

6. ## 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?

7. ## Re: Induction with derivatives

Originally Posted by zachd77
Sorry, this actually isn't a homework question. I was just curious. Sorry if I angered you slightly.
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.

8. ## 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

9. ## 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.

10. ## 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.