Proving the Product Rule for Differentiating x^n-1

[Perzik]

Homework Statement

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

Homework 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)

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

 

[radou]

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

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

 

[Perzik]

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

 

[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 x^n-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?

 

[radou]

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 x^n-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?

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.

 

[HallsofIvy]

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.