How to prove $D_x^k(x^k)=k!$?
Well, I know first I have to prove $D_x^1(x^1)=1!$ which is easy to prove, then I will prove
Assume $$D_x^k(x^k)=k!$$
Show: $$D_x^{k+1}(x^{k+1})=(k+1)!$$
Which I stuck on it.
2026-04-04 15:17:31.1775315851
How to prove $D_x^k(x^k)=k!$ using mathematical induction?
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$$ \begin{align} D_x^{k+1} x^{k+1} &= D_x^k D_x^1 x^{k+1} \\ &= (k+1) D_x^k x^k \end{align} $$
I think you got the rest