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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $f = a_0 + a_1 x + \\dots + a_n x^n \\in R[x]$.\nThe formal derivative of $f$ is the polynomial\n$$f' = a_1 + 2 a_2 x + 3 a_3 x^2 + \\dots + n a_n x^{n-1},$$\nwhere $k a_k$ is the sum of $k$ copies of $a_k$.\n\n## Remarks\n\nThe derivative is additive, and $(r f)' = r f'$ for each constant $r \\in R$.\nIf $R$ is commutative, the product rule holds: $(f g)' = f' g + f g'$.\nIn characteristic $p$, the derivative of $x^p$ is $p x^{p-1} = 0$. So a nonconstant polynomial can have derivative $0$.\nThe derivative detects [[repeated roots|067F8YVYMH]].\n\n## Explanations\n\n### L1 human\n\nThe formal derivative uses the rule from calculus, but it is pure algebra.\nYou never take a limit, so it works over the integers modulo $p$ too.\nFor example, the derivative of $x^3 + 2x$ is $3x^2 + 2$.\n",
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