# Proof of the Power Rule - Stuck at the End

1. Jan 20, 2013

### Astrum

1. The problem statement, all variables and given/known data
Prove that the following

$f'(x)=nx^{n-1}$ if $f(x)=x^{n}$

2. Relevant equations
Binomial theorem, definition of the derivative

3. The attempt at a solution

$$f'(x)=lim_{h\rightarrow0}\frac{f((x+h)^{n})-f(x)}{h}$$

We need to expand the (x+h)^2 term now

$$\sum^{n}_{k=0}{n\choose k} x^{n-k}h^{k}={n\choose 0}x^n+{n\choose 1}x^{n-1}h+...+{n\choose k}x^{n-k}h^{k}$$

So, we sub this for the f((x+h)^n) term:

$$lim_{h \rightarrow 0}\frac{{n\choose 0}x^n+{n\choose 1}x^{n-1}h+...+{n\choose k}x^{n-k}h^{k}-x}{h}$$

$${n\choose k}=\frac{n!}{(n-k)!k!}$$

This now simplifies to---

$$lim_{h\rightarrow0}\frac{x^{n}}{h}+nx^{n-1}-lim_{h\rightarrow0}\frac{x}{h}$$

The first and third terms create only a one sided limit, and they both go to infinity, I'm not sure where I went wrong....

I could just look up the proof, but I'm trying to do it by myself, so I'm only looking for a hint.

I feel like I'm really close, because I have the answer there, I just don't know why those other two terms are messing it up.

There are only two possibilities, either I've gone in the completely wrong direction, or I made a silly mistake somewhere.

Last edited: Jan 20, 2013
2. Jan 20, 2013

### SteamKing

Staff Emeritus
Your definition of f'(x) is a little strange.

Since f(x) = x^n, then

f'(x) = lim (h -> 0) [(x+h)^n - x^n) / h]

Try expanding now using the Binomial Theorem.

3. Jan 20, 2013

### Astrum

Thanks, I didn't catch that.... how silly.

So, that means those two terms cancel out, and we're left with the correct one.

4. Jan 20, 2013

### LCKurtz

What do you mean by "we're left with the correct one"? There are two terms that do cancel but there are a lot of others and you have to deal with the h's. Have you actually worked it out?

5. Jan 20, 2013

### Astrum

Yes, I skipped steps out of laziness (meaning, I did them on paper, but I didn't feel like taking the time to write them out here).

The h on the bottom cancels out and all the hs on the top go to 0.