Prove that $d^n(x^n)/dx^n = n!$ by induction I need to prove that $d^n(x^n)/dx^n = n!$ by induction.
Any help?

Let's start the induction at $n=1$ (could also be done at $n=0$ but that might be a bit confusing).
$$\dfrac{d^1}{dx^1}x^1 = \dfrac{d}{dx}x = 1 = 1! $$
Now let's try at any $n>1$:
$$\dfrac{d^n}{dx^n}x^n = \dfrac{d^{n-1}}{dx^{n-1}} \lbrace \dfrac{d}{dx}x^n \rbrace = \dfrac{d^{n-1}}{dx^{n-1}} \lbrace nx^{n-1} \rbrace $$
move the constant out of the bracket, and then use the induction step (for $n-1$):
$$ =n(n-1)!=n! $$
