How can we show this by using mathematical induction
(d^n\dx^n )x^n = n!
First step . We have
$\displaystyle \dfrac{d^1}{dx^1}x^1=1=1!$
so, the equality is true for $\displaystyle n=1$
What difficulties have you had in the second step?
Remark . If the question is for $\displaystyle n\geq 0$ then, take into account that:
$\displaystyle \dfrac{d^0}{dx^0}f(x)=f(x)$ and $\displaystyle 0!=1$
Fernando Revilla
Suppose:
$\displaystyle \dfrac{d}{dx^n}x^n=n!$
Then, prove:
$\displaystyle \dfrac{d}{dx^{n+1}}x^{n+1}=(n+1)!$
Fernando Revilla