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  "body": "\n## Summary\n\nThe Pincherle derivative of a linear operator $T$ on polynomials is the commutator $T' = Tx - xT$ of $T$ with multiplication by $x$.\nIt behaves like a derivative: it is linear and satisfies the product rule $(ST)' = S'T + ST'$.\nIt turns the [[differentiation operator|BB15T3J891]] into the identity, and each shift operator $f(x) \\mapsto f(x + h)$ into $h$ times itself.\nIt is a basic tool in umbral calculus.\n\n## Explanations\n\n### L1 human\n\nOperators on polynomials can be \"differentiated\" too.\nThe Pincherle derivative measures how far an operator is from commuting with multiplication by $x$.\nOperators that commute with multiplication by $x$, such as multiplication by a fixed polynomial, have derivative $0$, like constants.\nDifferentiation $d/dx$ has Pincherle derivative the identity, just as $x$ has derivative $1$.\n",
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