Remark 1

Tag 0853, permanent. Cite it as [Bourbaki, Tag 0853].

Functions of a Real Variable, chapter I, § 3. DERIVATIVES OF HIGHER ORDER

We have already noticed in the proof of th. 1 that if $f$ has a derivative of order $n$ on I, and if

f(x) = a_0 + a_1(x - a) + a_2(x - a)^2 + \cdots + a_n(x - a)^n + r_n(x)

is its Taylor expansion of order

n

at the point

a

, then the Taylor expansion of order

n - 1

for

f'

at the point

a

f'(x) = a_1 + 2a_2(x - a) + \cdots + na_n(x - a)^{n-1} + r'_n(x).

We say that it is obtained from the expansion (11) of $f$ by differentiating term-by-term.

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

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