Finding Jordan form of differentiation map

Question

Consider the space $V$ spanned over $\mathbb{R}$ by $\{1,x,e^x,xe^x,x^2e^x,x^3e^x\}$ and let $D:V \rightarrow V$ be the differentiation map. I need to show that $D$ maps $V \rightarrow V$ and find the Jordan form of $D$ on $V$.
To show it spans, I need to show $im(D)$ contains all basis vectors of $V$. We have: $$D(1)=0$$ $$D(x)=1$$ $$D(e^x)=e^x$$ $$D(xe^x)=e^x+xe^x$$ $$D(x^2e^x)=2xe^x+x^2e^x$$ $$D(x^3e^x)=3x^2e^x+x^3e^x$$ By considering linear combinations of these, I can see that $im(D)$ contains $1,e^x,xe^x,x^2e^x,x^3e^x$, but it does not contain $x$. I am confused about this. Also, how do I find the Jordan form of $D$?