Let $D: F[t]_{n} \rightarrow F[t]_{n}$ be the differential map. Let $f \in (F[t]_{n})^{*}$ be the functional assigning to each $y(t) \in F[t]_{n}$ the integral $\int_{0}^{1}y(t) dt$. Compute the functional $D^{*}(f)$.
In general we have that given a linear map $T:V\rightarrow W$ where $V,W$ are subspaces, the we get the dual map defined by:
$T^{*}: W^{*} \rightarrow V^{*}$ then $(T^{*}(f))(v) = f(Tv)$
So in this case we will have for a polynomial $p = a_{n}t^{n}+...+a_{0}$:
$(D^{*}(f))(p) = f(Dp) = f(D(p = a_{n}t^{n}+...+a_{0})) = a_{n}+ a_{n-1}+...a_{1}$
Am I right?
Assuming you meant this: $$f(D(p)) = f(na_nt^{n-1} + (n-1)a_{n-1}t^{n-2} + \cdots + a_1) = a_n + a_{n-1} + \cdots + a_1$$ then yes, this is correct.