Example 2

Tag 083A, permanent. Cite it as [Bourbaki, Tag 083A].

Functions of a Real Variable, chapter I, § 1. FIRST DERIVATIVE

The real function $x^n$ arises from the multilinear function

(x_1, x_2, \ldots, x_n) \mapsto x_1 x_2 \ldots x_n

defined on $\mathbf{R}^n$ , by substituting $x$ for each of the $x_i$ ; so prop. 3 shows that $x^n$ is differentiable on $\mathbf{R}$ and has derivative $nx^{n-1}$ . As a result the polynomial function $\mathbf{a}_0 x^n + \mathbf{a}_1 x^{n-1} + \cdots + \mathbf{a}_{n-1} x + \mathbf{a}_n$ (the $\mathbf{a}_i$ being constant vectors) has derivative

n \mathbf{a}_0 x^{n-1} + (n-1) \mathbf{a}_1 x^{n-2} + \cdots + \mathbf{a}_{n-1};

when the $\mathbf{a}_i$ are real numbers this function coincides with the derivative of a polynomial function as defined in Algebra (A, IV).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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