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   143 

   144 <h1 class="title">An Unambiguous Notation for Derivatives</h1>

   145 

   146 

   147 

   148 

   149 

   150 

   151 

   152 <div id="table-of-contents">

   153 <h2>Table of Contents</h2>

   154 <div id="text-table-of-contents">

   155 <ul>

   156 <li><a href="#sec-1">1 Calculus of Infinitesimals </a>

   157 <ul>

   158 <li><a href="#sec-1-1">1.1 Differential Objects </a></li>

   159 <li><a href="#sec-1-2">1.2 Interactions obey the chain rule </a></li>

   160 </ul>

   161 </li>

   162 </ul>

   163 </div>

   164 </div>

   165 

   166 <div id="outline-container-1" class="outline-2">

   167 <h2 id="sec-1"><span class="section-number-2">1</span> Calculus of Infinitesimals </h2>

   168 <div class="outline-text-2" id="text-1">

   169 

   170 

   171 </div>

   172 

   173 <div id="outline-container-1-1" class="outline-3">

   174 <h3 id="sec-1-1"><span class="section-number-3">1.1</span> Differential Objects </h3>

   175 <div class="outline-text-3" id="text-1-1">

   176 

   177 

   178 <p>

   179 A <b>differential object</b> is a pair \([x,\,dx]\) consisting of a variable

   180 and an infinitely small increment of it. Differential objects can

   181 interact with functions, producing a new differential object as a

   182 result; this interaction is for calculating derivatives of functions.

   183 </p>

   184 <p>

   185 Differential objects are for

   186 calculating derivatives of functions: the derivative of \(f\) with

   187 respect to \(x\) 

   188 </p>

   189 <p>

   190 You can &ldquo;apply&rdquo;

   191 functions to differential objects; the result is:

   192 </p>

   193 

   194 

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

   196 

   197 <p>

   198 Loosely speaking, the interaction of \(f\) and a differential object

   199 of \(x\) is a differential object of \(f\).

   200 </p>

   201 

   202 </div>

   203 

   204 </div>

   205 

   206 <div id="outline-container-1-2" class="outline-3">

   207 <h3 id="sec-1-2"><span class="section-number-3">1.2</span> Interactions obey the chain rule </h3>

   208 <div class="outline-text-3" id="text-1-2">

   209 

   210 

   211 <p>

   212 The interaction of \(f\) and the differential object \([x, dx]\) is

   213 a differential object \([f(x), Df(x)\cdot dx]\). Because of the rule for

   214 interactions, if you apply another function \(g\), you get the

   215 chain-rule answer you expect:

   216 </p>

   217 

   218 

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

   220 Dg(f(x))\cdot Df(x)\cdot dx\right]\)

   221 

   222 

   223 

   224 

   225 

   226 

   227 

   228 

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   231 </div>

   232 <div id="postamble">

   233 <p class="date">Date: 2011-08-08 02:49:24 EDT</p>

   234 <p class="author">Author: Dylan Holmes</p>

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author	Robert McIntyre <rlm@mit.edu>
date	Fri, 28 Oct 2011 00:03:05 -0700
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