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rlm@2: rlm@2:An Unambiguous Notation for Derivatives
rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2:Table of Contents
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- 1 Calculus of Infinitesimals
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- 1.1 Differential Objects rlm@2:
- 1.2 Interactions obey the chain rule rlm@2:
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1 Calculus of Infinitesimals
rlm@2:1.1 Differential Objects
rlm@2:rlm@2: A differential object is a pair \([x,\,dx]\) consisting of a variable rlm@2: and an infinitely small increment of it. Differential objects can rlm@2: interact with functions, producing a new differential object as a rlm@2: result; this interaction is for calculating derivatives of functions. rlm@2:
rlm@2:rlm@2: Differential objects are for rlm@2: calculating derivatives of functions: the derivative of \(f\) with rlm@2: respect to \(x\) rlm@2:
rlm@2:rlm@2: You can “apply” rlm@2: functions to differential objects; the result is: rlm@2:
rlm@2: rlm@2: rlm@2: \([x,dx]\xrightarrow{\quad f \quad}[f(x), Df(x)\cdot dx].\) rlm@2: rlm@2:rlm@2: Loosely speaking, the interaction of \(f\) and a differential object rlm@2: of \(x\) is a differential object of \(f\). rlm@2:
rlm@2: rlm@2:1.2 Interactions obey the chain rule
rlm@2:rlm@2: The interaction of \(f\) and the differential object \([x, dx]\) is rlm@2: a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for rlm@2: interactions, if you apply another function \(g\), you get the rlm@2: chain-rule answer you expect: rlm@2:
rlm@2: rlm@2: rlm@2: \([f(x), Df(x)\cdot dx]\xrightarrow{\quad g\quad}\left[g(f(x)),\, rlm@2: Dg(f(x))\cdot Df(x)\cdot dx\right]\) rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2: rlm@2:Date: 2011-08-08 02:49:24 EDT
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