What is the rank of the differentiation operator on Pn? What is the kernel?

Question

For this question I was thinking of saying $P_n(x)=Ax^n+Ax^{n-1}+...+Ax^2+Ax^1+1$ and finding the first derivative $P'n(x)=n.Ax^{n-1}+(n-1)Ax^{n-2}+...+2Ax^1+A+0$ so in matrix form would get a $n$ by $1$ matrix with $n-1$ rows being non-zero and $1$ row being zero. So the rank is $n-1$ and form the rank-nullity theorem $Rank(A)+N(A)=n$ where $N(A)=dim (Ker(Pn))=1$. I am unsure if this is correct as well if the differentiation operator just means to differentiation one time or to differentiation up to n times. Also unsure if the constants A should have some index to differentiation them and if its $ker(P_n)$ and not $Ker(A)$.

Answer 1

Consider $\dim P_n=n+1$ and clearly $\dim\ker(A)=1$ hence $\dim\operatorname{im}(A)=n$