for every $x \in \mathbb{R}$. Then $p$ is continuous on $\mathbb{R}$, differentiable on $\mathbb{R}$, and its derivative $p': \mathbb{R} \to \mathbb{R}$ is given by
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\begin{align*}
p'(x)=\sum_{i=1}^{d} i a_i x^{i-1}
\end{align*}
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for every $x \in \mathbb{R}$, where the sum is interpreted as $0$ when $d=0$.
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Moreover, the function $P: \mathbb{R} \to \mathbb{R}$ defined by