Let $F$ be a field of characteristic zero and let $D$ be the formal polynomial differentiation map so that $$D(a_0+a_1x+a_2x^2+....+a_nx^n)=a_1+2a_2x+3a_3x^2+....+na_nx^{n-1}$$ Find the image of $F[x]$ under $D$.
The answer is that the image of $F[x]$ under $D$ is $F[x]$, Im$(D)$
I understand as far as Im$(D)<F[x]$ (i.e., Im$(D)$ is a subring of $F[x]$) but I don't see why
Im$(D)=F[x]$. Can you guys please help?
Hint: $$ D\Bigl(\frac{x^{n+1}}{n+1}\Bigr)= $$
Further hint: as soon as you have proved that $x^n\in\operatorname{Im}(D)$, for every $n\ge0$, you can apply the fact that $D$ is linear.