# Product Rule

1. Dec 3, 2006

### Perzik

1. The problem statement, all variables and given/known data

Use the product rule to show that dx^n-1/dx = (n-1)x^n-2

2. Relevant equations

The general idea is..

If: h(x) = f(x)g(x)
Then: dh(x)/dx = f(x)dg(x)/dx + g(x)df(x)/d(x)

3. The attempt at a solution

It seems like a simple solution but everytime I attempt solving it I get confused and end up with a totally wrong answer

2. Dec 3, 2006

Define f(x) and g(x) and everything will be easy.

Hint: $$a^n\cdot a^m = a^{n+m}$$

3. Dec 3, 2006

### Perzik

that's the problem I'm getting..I'm not sure where to start to determine f(x) and g(x).

4. Dec 3, 2006

### StatusX

I don't understand this question. What's with the n-1? I'm assuming you don't know the derivative of xn, or at least you're not allowed to use it here, since otherwise you could just replace n with n-1 and get the answer immediately. I guess they want you to derive from scratch what the derivative of xn-1 is. Note (as above) this is equivalent to deriving what the derivative of xn is, which is much less silly. Do you know how to use induction, and can you think of how to apply it here?

5. Dec 4, 2006

Hm, since there's the product rule 'involved', I'd assume one can use the knowledge of what the derivative of x^n equals, but Perzik should know best what the problem asks for.

6. Dec 4, 2006

### HallsofIvy

Staff Emeritus
The point is to use "proof by induction".

If n= 1, what is $dx/dx$? Does that match the formula? What is (1)x0?

Now assume that, for some k,$dx^k/dx= k x^{k-1}$. Write $x^{k+1}$ as $x(x^k)$ and use the product rule.