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The formal derivative $D (\sum a_{n} x^{n}) = \sum_{n \geq 1} n a_{n} x^{n - 1}$

Definition

For $f = \sum_{n \geq 0} a_{n} x^{n} \in R ⟦ x ⟧$ , its formal derivative is

$D f = f^{'} : = \sum_{n \geq 1} n a_{n} x^{n - 1},$

where $n a_{n}$ means the sum of $n$ copies of $a_{n}$ in the additive group of $R$ . Define $D^{0} f = f$ and $D^{k + 1} f = D (D^{k} f)$ .

This is coefficientwise algebra, not a limit. In particular $D (x^{n}) = n x^{n - 1}$ for $n \geq 1$ and $D (1) = 0$ . In a ring of characteristic $p > 0$ , one can have $D (x^{p}) = 0$ although $x^{p}$ is not constant.

Depends on

Used by

Formal differentiation is linear and satisfies product, power, quotient, chain, and coefficient-recovery laws Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Benjamin Sambale, An Invitation to Formal Power Series (standard reference, not scraped)

The formal derivative D(∑anxn)=∑n≥1nanxn−1D(∑an​xn)=∑n≥1​nan​xn−1