Derivation operator/Polynomials of maximal degree/Nilpotent/Exercise

Let

$D\colon \mathbb {R} [X]_{\geq m}\longrightarrow \mathbb {R} [X]_{\geq m}$

be the restriction of the derivation operator ${}P\mapsto P'$ to the space of polynomials of degree ${}\leq m$. Show that ${}D$ is nilpotent. Show also that

$D\colon \mathbb {R} [X]\longrightarrow \mathbb {R} [X]$

is not nilpotent.

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