---
tag: BB15T3J891
kind: concept
name: Differentiation operator p
msc: [44A45, 44A40]
sources:
  - url: https://en.wikipedia.org/wiki/Operational_calculus
---

## Summary

In [[operational calculus|8050R3Z6HA]], $p$ denotes the operator $d/dt$ that sends a function of time to its [[derivative|0V1EJCT8G6]].
It is treated as an algebraic symbol: one forms polynomials and fractions in $p$, and $1/p$ is integration from $0$.
Because $p$ undoes integration but integration does not quite undo $p$, initial values appear in the rules, as in $p^{-1} p f = f - f(0)$.

## Explanations

### L1 human

Writing $p$ for "take the derivative" lets one write $y'' + 3y' + 2y$ as $(p^2 + 3p + 2) y$, and even factor it as $(p + 1)(p + 2) y$.
Its partner $1/p$ means "integrate from $0$".
Differentiating an integral gives back the function, but integrating a derivative from $0$ loses the starting value.
For example, for $f(t) = t + 5$, integrating $f' = 1$ from $0$ gives $t$, not $t + 5$.