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?