Formal derivative

Definition

Let $R$ be a ring and let $f = a_0 + a_1 x + \dots + a_n x^n \in R[x]$ . The formal derivative of $f$ is the polynomial $f' = a_1 + 2 a_2 x + 3 a_3 x^2 + \dots + n a_n x^{n-1},$ where $k a_k$ is the sum of $k$ copies of $a_k$ .

Remarks

The derivative is additive, and $(r f)' = r f'$ for each constant $r \in R$ . If $R$ is commutative, the product rule holds: $(f g)' = f' g + f g'$ . In characteristic $p$ , the derivative of $x^p$ is $p x^{p-1} = 0$ . So a nonconstant polynomial can have derivative $0$ . The derivative detects repeated roots.

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Group concept
Ring definition
Polynomial ring definition
Formal derivative definition

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36 pages build on this one.

Tag	`0AD4VS02H3`
URI	`m2://0AD4VS02H3`
Kind	definition
Writing	W0 stub
MSC	13N15
Defines	Formal derivative
Uses	Polynomial ring

Formal derivative

Definition

Remarks

Reading path

Where this leads

Further reading

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