= Solution
For affine <polynomials> $f(t)=a+bt$ and $g(t)=c+dt$, direct integration gives $\langle f,g\rangle=2ac+2bd/3$. This is bilinear and symmetric, and $\langle f,f\rangle=2a^2+2b^2/3>0$ for every nonzero $f$. Thus it is an <inner product>.
The <adjoint of differentiation on affine polynomials> is determined by $\langle Df,g\rangle=\langle f,D^*g\rangle$. Since $Df=b$, the left side is $2bc$. Writing $D^*g=A+Bt$, the right side is $2aA+2bB/3$. Equality for every $a,b$ forces $A=0$, $B=3c$, hence
$$
\boxed{D^*(c+dt)=3ct.}
$$
This is not simply negative differentiation: there are no endpoint conditions here that would remove the boundary term in an <integration by parts> argument.
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