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  "body": "\n## Definition\n\nLet $K$ be a field and let $K[x]$ be the [[polynomial ring|0SDC934R52]] in one variable over $K$.\nLet $T \\colon K[x] \\to K[x]$ be a $K$-[[linear map|0NRYHJPQG2]].\nThe *Pincherle derivative* of $T$ is the linear map $T'$ on $K[x]$ given by\n$$T'(f) = T(x f) - x \\, T(f) \\qquad (f \\in K[x]).$$\nWriting $x$ also for the operator of multiplication by $x$, this says $T' = T x - x T$.\n\n## Remarks\n\nThe map $T \\mapsto T'$ is linear and satisfies the product rule $(ST)' = S'T + ST'$, since $STx - xST = S(Tx - xT) + (Sx - xS)T$.\n\nExample: for $D = d/dx$, $D'(f) = (xf)' - x f' = f$, so $D' = \\id$.\nFor the shift $E^h f(x) = f(x + h)$ with $h \\in K$, $(E^h)'(f)(x) = (x + h) f(x + h) - x f(x + h) = h f(x + h)$, so $(E^h)' = h E^h$.\n\nOperators with Pincherle derivative zero: a multiplication operator $M_q f = q f$ with $q \\in K[x]$ commutes with $x$, so $M_q' = 0$.\nConversely, $T' = 0$ forces $T(x^n) = x^n T(1)$ for all $n$, so $T = M_{T(1)}$.\n",
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