LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

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Repeated derivatives along a line expand by the multinomial formula

Statement

Let $k \in N$ , let $U \subseteq R^{m}$ be open, $f \in C^{k} (U)$ , and let $I \subseteq R$ be an open interval such that $a + t h \in U$ for every $t \in I$ . Write $ι : N \to R$ for the canonical-natural map of The canonical natural $ι (n) = n \cdot 1_{F}$ of a field. For $g (t) = f (a + t h)$ and every $0 \leq r \leq k$ ,

$g^{(r)} (t) = \sum_{∣ α ∣ = r} \frac{ι (r!)}{ι (α!)} D^{α} f (a + t h) h^{α} (t \in I) .$

Facts & Assumptions

Given: The stated open-domain, open-interval, $C^{k}$ , and direction hypotheses.

[L1]

A function with continuous first partial derivatives near a point is totally differentiable there, and the total chain rule then applies to the affine line map $t \mapsto a + t h$ (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[L2]

Ordered mixed derivatives through order $k$ commute under permutation (Continuous mixed partials of order $k$ are invariant under permutations).

[L3]

The multi-index conventions $∣ α ∣$ , $α!$ , $h^{α}$ , and the canonical derivative $D^{α} f$ are those of $C^{k}$ maps and multi-index derivative notation in Euclidean space.

[L4]

The canonical-natural map carries finite natural sums and products to the corresponding real sums and products (The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Laws of finite sums and products in $N$ , and $ι (\sum_{k < n} a_{k}) = \sum_{k < n} ι (a_{k})$ ).

Proof

technique · induction

1.1

For $r = 0$ the displayed sum consists of the zero multi-index and equals $f (a + t h) = g (t)$ .

baseL3

1.2

Fix $r < k$ and assume the formula at order $r$ .

1.3

Each $D^{α} f$ with $∣ α ∣ = r$ has continuous first partials, so [L1] differentiates its composition with the affine line. By [L3], we use the canonical multi-index notation for the resulting derivatives. When $r = 0$ the resulting first derivatives are already canonical; when $r \geq 1$ , [L2] permits the resulting derivatives to be written as $D^{α + e_{i}} f$ . By [L4], collecting the coefficient of a fixed $β$ with $∣ β ∣ = r + 1$ gives

$\sum_{i : β_{i} > 0} \frac{ι (r!)}{ι ((β - e_{i})!)} = \frac{ι (r!)}{ι (β!)} \sum_{i < m} ι (β_{i}) = \frac{ι ((r + 1)!)}{ι (β!)} .$

Thus the formula at order $r + 1$ follows. [step 1.2, L1, L2, L3, L4, algebra]

2.1

Steps 1.1--2.1 prove the formula successively for every $r \leq k$ .

step 1.1step 1.2step 1.3discharge-induction∎

Depends on

Used by

Multivariable Taylor formula with a Lagrange remainder along a line segment Theorem

Dependency tree · two levels

44 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MAT237 notes: Taylor's theorem in several variables (standard reference, not scraped)