(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Sorry I can't do this in latex, the PC I'm on doesn't display it and I can't confidently reel it off without checking so I'll have to do this in an ugly way! :/

Prove:

d^n/dx^n [(x-1)^n] = n!

For integer n

2. Relevant equations

I'm trying induction but it's never been my strong point, I'm not convinced I actually understand how induction works, whenever I try and get my head around it, it seems circular..

3. The attempt at a solution

Testing when n=0 gives (x-1)^0 = 1 = 0! so it holds when n=0

Now assuming true for n=k gives:

d^k/dx^k [(x-1)^k] = k!

If this holds for n=k, that it holds when n=k+1 must be implied by n=k being true:

d^(k+1)/dx^(k+1) [x-1]^(k+1) = d^k/dx^k(d/dx((x-1)^(k+1))) separating the differentiation operator

= d^k/dx^k((k+1)(x-1)^k) evaluating the inner derivative

= (k+1)d^k/dx^k((x-1)^k) bringing the constant (k+1) outside the derivative

= (k+1) k! using the result I'm trying to prove

= (k+1)!

Sorry, that's hideous.. is it OK?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proving d^l[(x-1)^l]/dx^l = l!

**Physics Forums | Science Articles, Homework Help, Discussion**