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.

Explanations

L1 human

The formal derivative uses the rule from calculus, but it is pure algebra. You never take a limit, so it works over the integers modulo $p$ too. For example, the derivative of $x^3 + 2x$ is $3x^2 + 2$.