linear algebra - Adjoint of derivative operator on polynomial space

I was working on a problem when I made the following reasoning.

I know that every linear operator $T:V \longrightarrow V$ on a Hilbert space $(V,\langle.,.\rangle)$ such that $\dim(V)<\infty$ has one (unique) adjoint operator $T^*:V \longrightarrow V$ (that is, $\langle T u,v\rangle = \langle u, T^* v \rangle$ $\forall u,v \in V$).

So if $V:=P_n$ is the space of all polynomials with degree less than or equal to $n \in \mathbb{N}$ (which gives $\dim(V)=n+1<\infty$) and $\langle f,g \rangle := \int_0^1f(t)g(t) \, dt$, what is the adjoint of the derivative operator $T=\dfrac{d}{dt}$?

I've tried to solve that, but still to no avail. I wonder if that is a silly question, but I haven't had any success searching for the answer either, so I apologize in advance if that's the case.