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calculus1.html
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<link rel="stylesheet" href="https://openmath.org/public/css/hyde.css"/>
<link rel="stylesheet" href="https://openmath.org/public/css/main.css"/>
<link rel="shortcut icon" sizes="144x144" type="image/png" href="/public/favicon.png"/>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_SVG"> </script>
<script type="text/javascript"> function divfold(_Id){
var thisLevel = document.getElementById(_Id);
var thisLevela = document.getElementById(_Id.concat("a"))
if(thisLevel.style.display != "none"){
thisLevel.style.display = "none";
thisLevela.style.backgroundColor = "#CCCCCC";}
else{
thisLevel.style.display = "block";
thisLevela.style.backgroundColor = "#AAFFAA";}
}
</script>
<title>calculus1</title>
<link rel="stylesheet" href="https://openmath.org/cd/omcd.css" type="text/css"/>
</head>
<body class="theme-base-odk">
<div class="sidebar">
<div class="container">
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<h1>
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<img src="https://openmath.org/public/logo.png" alt="OpenMath" width="100%"/>
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</h1>
<p class="lead">OpenMath is an extensible standard for representing the semantics of mathematical objects.</p>
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</div>
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<div class="content container">
<div class="page">
<a name="top"/>
<h1>OpenMath Content Dictionary: calculus1</h1>
<dl>
<dt>
<span class="dt">Canonical URL:</span>
</dt>
<dd>
<a href="http://www.openmath.org/cd/calculus1.ocd">http://www.openmath.org/cd/calculus1.ocd</a>
</dd>
<dt>
<span class="dt">CD Base:</span>
</dt>
<dd>
<a href="http://www.openmath.org/cd">http://www.openmath.org/cd</a>
</dd>
<dt>
<span class="dt">CD File:</span>
</dt>
<dd>
<a href="calculus1.ocd">calculus1.ocd
</a>
</dd>
<dt>
<span class="dt">CD as XML Encoded OpenMath:</span>
</dt>
<dd>
<a href="calculus1.omcd">calculus1.omcd
</a>
</dd>
<dt>
<span class="dt">Defines:</span>
</dt>
<dd>
<a href="#defint">defint</a>, <a href="#diff">diff</a>, <a href="#int">int</a>, <a href="#nthdiff">nthdiff</a>, <a href="#partialdiff">partialdiff</a>, <a href="#partialdiffdegree">partialdiffdegree</a>
</dd>
<dt>
<span class="dt">Date:</span>
</dt>
<dd>2009-04-01</dd>
<dt>
<span class="dt">Version:</span>
</dt>
<dd>5</dd>
<dt>
<span class="dt">Review Date:</span>
</dt>
<dd>2014-04-01</dd>
<dt>
<span class="dt">Status:</span>
</dt>
<dd>official</dd>
</dl>
<hr/>
<pre>
This document is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
The copyright holder grants you permission to redistribute this
document freely as a verbatim copy. Furthermore, the copyright
holder permits you to develop any derived work from this document
provided that the following conditions are met.
a) The derived work acknowledges the fact that it is derived from
this document, and maintains a prominent reference in the
work to the original source.
b) The fact that the derived work is not the original OpenMath
document is stated prominently in the derived work. Moreover if
both this document and the derived work are Content Dictionaries
then the derived work must include a different CDName element,
chosen so that it cannot be confused with any works adopted by
the OpenMath Society. In particular, if there is a Content
Dictionary Group whose name is, for example, `math' containing
Content Dictionaries named `math1', `math2' etc., then you should
not name a derived Content Dictionary `mathN' where N is an integer.
However you are free to name it `private_mathN' or some such. This
is because the names `mathN' may be used by the OpenMath Society
for future extensions.
c) The derived work is distributed under terms that allow the
compilation of derived works, but keep paragraphs a) and b)
intact. The simplest way to do this is to distribute the derived
work under the OpenMath license, but this is not a requirement.
If you have questions about this license please contact the OpenMath
society at <a href="http://www.openmath.org">http://www.openmath.org</a>.
</pre>
<pre>
Author: OpenMath Consortium
SourceURL: <a href="https://github.com/OpenMath/CDs">https://github.com/OpenMath/CDs</a>
</pre>
<p>
This CD is intended to be compatible with the calculus operations
in Content MathML.</p>
<p> Integration is just for the univariate case and is either definite
or indefinite.
</p>
<hr/>
<h2>
<a name="diff">diff</a>
</h2>
<dl>
<dt>
<span class="dt">Role:</span>
</dt>
<dd>application</dd>
</dl>
<dl>
<dt>
<span class="dt">Description:</span>
</dt>
<dd>
<p>
This symbol is used to express ordinary differentiation of a unary
function. The single argument is the unary function.
</p>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Commented Mathematical property (CMP):</span>
</dt>
<dd>
diff(lambda y:a(y) + b(y))(x) = diff(lambda y:a(y))(x) + diff(lambda y:b(y))(x)
</dd>
</dl>
<dl>
<dt>
<span class="dt">Formal Mathematical property (FMP):</span>
</dt>
<dd>
<div>
<button id="id.om.15xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.15xml')">OpenMath XML (source)</button>
<pre id="id.om.15xml" style="display:none">
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
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<OMV name="y"/>
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<OMA>
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<OMV name="y"/>
</OMA>
</OMBIND>
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<OMV name="x"/>
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<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
</OMA>
</OMA>
</OMA>
</OMOBJ></pre>
</div>
<div>
<button id="id.om.15cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.15cmml')">Strict Content MathML</button>
<pre id="id.om.15cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><csymbol cd="arith1">plus</csymbol>
<apply><ci>a</ci><ci>y</ci></apply>
<apply><ci>b</ci><ci>y</ci></apply>
</apply>
</bind>
</apply>
<ci>x</ci>
</apply>
<apply><csymbol cd="arith1">plus</csymbol>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><ci>a</ci><ci>y</ci></apply>
</bind>
</apply>
<ci>x</ci>
</apply>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><ci>b</ci><ci>y</ci></apply>
</bind>
</apply>
<ci>x</ci>
</apply>
</apply>
</apply>
</math></pre>
</div>
<button id="id.om.15prefa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.15pref')">Prefix</button>
<div id="id.om.15pref" style="display:none; margin-top: 0.5em">
<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
(<a href="../../../cd/arith1.html#plus">plus</a>
( <i>a</i>
( <i>y</i>)
, <i>b</i>
( <i>y</i>)
)
)
)
( <i>x</i>)
, <a href="../../../cd/arith1.html#plus">plus</a>
(<a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
( <i>a</i>
( <i>y</i>)
)
)
( <i>x</i>)
, <a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
( <i>b</i>
( <i>y</i>)
)
)
( <i>x</i>)
)
)
</div>
<div>
<button id="id.om.15popa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.15pop')">Popcorn</button>
<div id="id.om.15pop" style="display:none; margin-top: 0.5em">
calculus1.diff(fns1.lambda[$y -> $a($y) + $b($y)])($x) = calculus1.diff(fns1.lambda[$y -> $a($y)])($x) + calculus1.diff(fns1.lambda[$y -> $b($y)])($x)
</div>
</div>
<div>
<button id="id.om.15mmla" style="width:20em; background-color:#AAFFAA"
onclick="divfold('id.om.15mml')">Rendered Presentation MathML</button>
<div id="id.om.15mml" style="display:block; margin-top: 0.5em">
<div style="overflow:auto;font-size:90%">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow xmlns:om="http://www.openmath.org/OpenMath">
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<mrow>
<mo>(</mo>
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<mi>d</mi>
<mrow>
<mi>d</mi>
<mrow>
<mi>y</mi>
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</mfrac>
<mrow>
<mo>(</mo>
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<mi>a</mi>
<mo>⁡</mo>
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<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
</div>
</div>
</div>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Commented Mathematical property (CMP):</span>
</dt>
<dd>
diff(lambda y:a(y) * b(y))(x) = diff(lambda y:a(y))(x) * b(x) +
a(x) * diff(lambda y:b(y))(x)
</dd>
</dl>
<dl>
<dt>
<span class="dt">Formal Mathematical property (FMP):</span>
</dt>
<dd>
<div>
<button id="id.om.60xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.60xml')">OpenMath XML (source)</button>
<pre id="id.om.60xml" style="display:none">
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
</OMA>
</OMA>
</OMBIND>
</OMA>
<OMV name="x"/>
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<OMA>
<OMS cd="arith1" name="plus"/>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
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<OMS cd="fns1" name="lambda"/>
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<OMV name="y"/>
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<OMA>
<OMV name="a"/>
<OMV name="y"/>
</OMA>
</OMBIND>
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<OMV name="x"/>
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<OMA>
<OMV name="b"/>
<OMV name="x"/>
</OMA>
</OMA>
<OMA>
<OMS cd="arith1" name="times"/>
<OMA>
<OMV name="a"/>
<OMV name="x"/>
</OMA>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="y"/>
</OMBVAR>
<OMA>
<OMV name="b"/>
<OMV name="y"/>
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<OMV name="x"/>
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</OMA>
</OMOBJ></pre>
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<div>
<button id="id.om.60cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.60cmml')">Strict Content MathML</button>
<pre id="id.om.60cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><csymbol cd="arith1">times</csymbol>
<apply><ci>a</ci><ci>y</ci></apply>
<apply><ci>b</ci><ci>y</ci></apply>
</apply>
</bind>
</apply>
<ci>x</ci>
</apply>
<apply><csymbol cd="arith1">plus</csymbol>
<apply><csymbol cd="arith1">times</csymbol>
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<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><ci>a</ci><ci>y</ci></apply>
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<ci>x</ci>
</apply>
<apply><ci>b</ci><ci>x</ci></apply>
</apply>
<apply><csymbol cd="arith1">times</csymbol>
<apply><ci>a</ci><ci>x</ci></apply>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>y</ci></bvar>
<apply><ci>b</ci><ci>y</ci></apply>
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</apply>
<ci>x</ci>
</apply>
</apply>
</apply>
</apply>
</math></pre>
</div>
<button id="id.om.60prefa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.60pref')">Prefix</button>
<div id="id.om.60pref" style="display:none; margin-top: 0.5em">
<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
(<a href="../../../cd/arith1.html#times">times</a>
( <i>a</i>
( <i>y</i>)
, <i>b</i>
( <i>y</i>)
)
)
)
( <i>x</i>)
, <a href="../../../cd/arith1.html#plus">plus</a>
(<a href="../../../cd/arith1.html#times">times</a>
(<a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
( <i>a</i>
( <i>y</i>)
)
)
( <i>x</i>)
, <i>b</i>
( <i>x</i>)
)
, <a href="../../../cd/arith1.html#times">times</a>
( <i>a</i>
( <i>x</i>)
, <a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>y</i>
] .
( <i>b</i>
( <i>y</i>)
)
)
( <i>x</i>)
)
)
)
</div>
<div>
<button id="id.om.60popa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.60pop')">Popcorn</button>
<div id="id.om.60pop" style="display:none; margin-top: 0.5em">
calculus1.diff(fns1.lambda[$y -> $a($y) * $b($y)])($x) = calculus1.diff(fns1.lambda[$y -> $a($y)])($x) * $b($x) + $a($x) * calculus1.diff(fns1.lambda[$y -> $b($y)])($x)
</div>
</div>
<div>
<button id="id.om.60mmla" style="width:20em; background-color:#AAFFAA"
onclick="divfold('id.om.60mml')">Rendered Presentation MathML</button>
<div id="id.om.60mml" style="display:block; margin-top: 0.5em">
<div style="overflow:auto;font-size:90%">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow xmlns:om="http://www.openmath.org/OpenMath">
<mrow>
<mrow>
<mo>(</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mrow>
<mi>y</mi>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mi>a</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>b</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mrow>
<mrow>
<mrow>
<mrow>
<mo>(</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mrow>
<mi>y</mi>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>b</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow>
<mrow>
<mi>a</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>⁢</mo>
<mrow>
<mrow>
<mo>(</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mrow>
<mi>y</mi>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</mrow>
</mrow>
</math>
</div>
</div>
</div>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Example:</span>
</dt>
<dd>
This represents the equation:
derivative(x + 1.0) = 1.0
<div>
<button id="id.om.114xmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.114xml')">OpenMath XML (source)</button>
<pre id="id.om.114xml" style="display:none"><OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
<OMS cd="relation1" name="eq"/>
<OMA>
<OMA>
<OMS cd="calculus1" name="diff"/>
<OMBIND>
<OMS cd="fns1" name="lambda"/>
<OMBVAR>
<OMV name="x"/>
</OMBVAR>
<OMA>
<OMS cd="arith1" name="plus"/>
<OMV name="x"/>
<OMF dec="1.0"/>
</OMA>
</OMBIND>
</OMA>
<OMV name="y"/>
</OMA>
<OMF dec="1.0"/>
</OMA>
</OMOBJ></pre>
</div>
<div>
<button id="id.om.114cmmla" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.114cmml')">Strict Content MathML</button>
<pre id="id.om.114cmml" style="display:none; margin-top: 0.5em">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply>
<apply><csymbol cd="calculus1">diff</csymbol>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="arith1">plus</csymbol><ci>x</ci><cn type="real">1.0</cn></apply>
</bind>
</apply>
<ci>y</ci>
</apply>
<cn type="real">1.0</cn>
</apply>
</math></pre>
</div>
<button id="id.om.114prefa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.114pref')">Prefix</button>
<div id="id.om.114pref" style="display:none; margin-top: 0.5em">
<a href="../../../cd/relation1.html#eq">eq</a>
(<a href="../../../cd/calculus1.html#diff">diff</a>
(<a href="../../../cd/fns1.html#lambda">lambda</a>
[
<i>x</i>
] .
(<a href="../../../cd/arith1.html#plus">plus</a>
( <i>x</i>, 1.0 )
)
)
( <i>y</i>)
, 1.0 )
</div>
<div>
<button id="id.om.114popa" style="width:20em; background-color:#CCCCCC"
onclick="divfold('id.om.114pop')">Popcorn</button>
<div id="id.om.114pop" style="display:none; margin-top: 0.5em">
calculus1.diff(fns1.lambda[$x -> $x + 1.0])($y) = 1.0
</div>
</div>
<div>
<button id="id.om.114mmla" style="width:20em; background-color:#AAFFAA"
onclick="divfold('id.om.114mml')">Rendered Presentation MathML</button>
<div id="id.om.114mml" style="display:block; margin-top: 0.5em">
<div style="overflow:auto;font-size:90%">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow xmlns:om="http://www.openmath.org/OpenMath">
<mrow>
<mrow>
<mo>(</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mi>d</mi>
<mrow>
<mi>x</mi>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.0</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>)</mo>
</mrow>
<mo>⁡</mo>
<mrow>
<mo>(</mo>
<mi>y</mi>
<mo>)</mo>
</mrow>
</mrow>
<mo>=</mo>
<mn>1.0</mn>
</mrow>
</math>
</div>
</div>
</div>
</dd>
</dl>
<dl>
<dt>
<span class="dt">Signatures:</span>
</dt>
<dd>
<a href="../../sts/calculus1.html#diff">