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Calculus/nth derivatives and induction

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Expert: - 10/17/2010


Question
1. The problem statement, all variables and given/known data
proving the nth derivative of x to the n power is n factorial


2. Relevant equations



3. The attempt at a solution
proving it for n=1
d^(1)x^1/dx = 1!=1 (a)
d/dx x^1 =1 (b)
a=b therefore at n=1 it is true
supposing it is true for n=k
then d^(k)x^k/dx = k!
verifying if it holds for n=k+1 and = (k+1)!

d^(k+1)x^(k+1)/dx = d/dx (d^(k)x^(k+1)/dx)
=d/dx(d^k/dx [x^k*x])=d/dx ([d^k/dx x^k * x]+[x^k d^k/dx x])
=d/dx ([k! * x]+[x^k * 0]
=d/dx x*k!=k!
this doesnt equal (k+1)! ,why?  

Answer
Sometimes it is better to just deal with the problem at the start.
It almost looks like the derivative order is being altered slightly,
but I will just give the straight answer.

If we have d^(k+1)x^(k+1)/dx^(k+1), take a derivative.
This gives (k+1)(d^k x/dx^k).

Since d^k x/dx^k = k!, the last equation is (k+1)k! = (k+1)!.

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