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An Unambiguous Notation for Derivatives

Table of Contents

1 Calculus of Infinitesimals

1.1 Differential Objects

A differential object is a pair \([x,\,dx]\) consisting of a variable and an infinitely small increment of it. Differential objects can interact with functions, producing a new differential object as a result; this interaction is for calculating derivatives of functions.

Differential objects are for calculating derivatives of functions: the derivative of \(f\) with respect to \(x\)

You can “apply” functions to differential objects; the result is:

\([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\)

Loosely speaking, the interaction of \(f\) and a differential object of \(x\) is a differential object of \(f\).

1.2 Interactions obey the chain rule

The interaction of \(f\) and the differential object \([x, dx]\) is a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for interactions, if you apply another function \(g\), you get the chain-rule answer you expect:

\([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\, Dg(f(x))\cdot Df(x)\cdot dx\right]\)

Date: 2011-08-08 02:49:24 EDT

Author: Dylan Holmes

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