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<option>---..acos</option>
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<center><b><big><big>AD Standard Math Unary Functions</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>



<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Evaluates the one argument standard math function 

<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
 where its argument is an 
<a href="glossary.xml#AD of Base" target="_top"><span style='white-space: nowrap'>AD&#xA0;of</span></a>
 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
 object.

<br/>
<br/>
<b><big><a name="x" id="x">x</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
 has one of the following prototypes

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;VecAD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;::reference&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
<br/>
<b><big><a name="y" id="y">y</a></big></b>
<br/>
The result 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>y</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
<br/>
<b><big><a name="Operation Sequence" id="Operation Sequence">Operation Sequence</a></big></b>
<br/>
Most of these functions are AD of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>

<a href="glossary.xml#Operation.Atomic" target="_top"><span style='white-space: nowrap'>atomic&#xA0;operations</span></a>
.
In all cases,
The AD of 
<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>

operation sequence used to calculate 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 is 
<a href="glossary.xml#Operation.Independent" target="_top"><span style='white-space: nowrap'>independent</span></a>

of 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="fun" id="fun">fun</a></big></b>
<br/>
A definition of 
<code><i><font color="black"><span style='white-space: nowrap'>fun</span></font></i></code>
 is included 
for each of the following functions:
<code><font color="blue">acos</font></code>,
<code><font color="blue">asin</font></code>,
<code><font color="blue">atan</font></code>,
<code><font color="blue">cos</font></code>,
<code><font color="blue">cosh</font></code>,
<code><font color="blue">exp</font></code>,
<code><font color="blue">fabs</font></code>,
<code><font color="blue">log</font></code>,
<code><font color="blue">log10</font></code>,
<code><font color="blue">sin</font></code>,
<code><font color="blue">sinh</font></code>,
<code><font color="blue">sqrt</font></code>,
<code><font color="blue">tan</font></code>,
<code><font color="blue">tanh</font></code>.


<br/>
<br/>
<b><big><a name="Examples" id="Examples">Examples</a></big></b>
<br/>
The following files
contain examples and tests of these functions.   
Each test returns true if it succeeds and false otherwise.

<table><tr><td align='left'  valign='top'>

<a href="abs.cpp.xml" target="_top">abs.cpp</a></td><td>
AD Absolute Value Function: Example and Test</td></tr>
<tr><td>

<a href="acos.cpp.xml" target="_top">Acos.cpp</a></td><td>
The AD acos Function: Example and Test</td></tr>
<tr><td>

<a href="asin.cpp.xml" target="_top">Asin.cpp</a></td><td>
The AD asin Function: Example and Test</td></tr>
<tr><td>

<a href="atan.cpp.xml" target="_top">atan.cpp</a></td><td>
The AD atan Function: Example and Test</td></tr>
<tr><td>

<a href="cos.cpp.xml" target="_top">cos.cpp</a></td><td>
The AD cos Function: Example and Test</td></tr>
<tr><td>

<a href="cosh.cpp.xml" target="_top">cosh.cpp</a></td><td>
The AD cosh Function: Example and Test</td></tr>
<tr><td>

<a href="exp.cpp.xml" target="_top">exp.cpp</a></td><td>
The AD exp Function: Example and Test</td></tr>
<tr><td>

<a href="log.cpp.xml" target="_top">log.cpp</a></td><td>
The AD log Function: Example and Test</td></tr>
<tr><td>

<a href="log10.cpp.xml" target="_top">log10.cpp</a></td><td>
The AD log10 Function: Example and Test</td></tr>
<tr><td>

<a href="sin.cpp.xml" target="_top">sin.cpp</a></td><td>
The AD sin Function: Example and Test</td></tr>
<tr><td>

<a href="sinh.cpp.xml" target="_top">sinh.cpp</a></td><td>
The AD sinh Function: Example and Test</td></tr>
<tr><td>

<a href="sqrt.cpp.xml" target="_top">sqrt.cpp</a></td><td>
The AD sqrt Function: Example and Test</td></tr>
<tr><td>

<a href="tan.cpp.xml" target="_top">tan.cpp</a></td><td>
The AD tan Function: Example and Test</td></tr>
<tr><td>

<a href="tanh.cpp.xml" target="_top">tanh.cpp</a></td><td>
The AD tanh Function: Example and Test</td></tr>
<tr><td>

</td></tr>
</table>
<br/>
<b><big><a name="Derivatives" id="Derivatives">Derivatives</a></big></b>
<br/>
Each of these functions satisfy a standard math function differential equation.
Calculating derivatives using this differential equation 
is discussed for 
both <a href="forwardtheory.xml#Standard Math Functions" target="_top"><span style='white-space: nowrap'>forward</span></a>

and <a href="reversetheory.xml#Standard Math Functions" target="_top"><span style='white-space: nowrap'>reverse</span></a>
 mode.
The exact form of the differential equation
for each of these functions is listed below:

<br/>
<br/>
<b><a name="Derivatives.acos" id="Derivatives.acos">acos</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>acos</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mo stretchy="false">-</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.asin" id="Derivatives.asin">asin</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>asin</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mrow><mn>-1</mn>
<mo stretchy="false">/</mo>
<mn>2</mn>
</mrow>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.atan" id="Derivatives.atan">atan</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>atan</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>1</mn>
<mo stretchy="false">+</mo>
<msup><mi mathvariant='italic'>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.cos" id="Derivatives.cos">cos</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>cos</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mo stretchy="false">-</mo>
<mi>sin</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>sin</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>cos</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.cosh" id="Derivatives.cosh">cosh</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>cosh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>sinh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>sin</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>cosh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.exp" id="Derivatives.exp">exp</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>exp</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.log" id="Derivatives.log">log</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>log</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.log10" id="Derivatives.log10">log10</a></b>
<br/>
This function is special in that it's derivatives are calculated
using the relation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>log</mi>
<mn>10</mn>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>log</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">/</mo>
<mi>log</mi>
<mo stretchy="false">(</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.sin" id="Derivatives.sin">sin</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>sin</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>cos</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>cos</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mo stretchy="false">-</mo>
<mi>sin</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.sinh" id="Derivatives.sinh">sinh</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>sinh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>cosh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>cosh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mi>sinh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.sqrt" id="Derivatives.sqrt">sqrt</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>sqrt</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mfrac><mrow><mn>1</mn>
</mrow>
<mrow><mn>2</mn>
<mrow><mstyle mathvariant='normal'><mi mathvariant='normal'>sqrt</mi>
</mstyle></mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.tan" id="Derivatives.tan">tan</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>tan</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mn>1</mn>
<mo stretchy="false">+</mo>
<mi>tan</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mtd></mtr></mtable>
</mrow></math>

<br/>
<b><a name="Derivatives.tanh" id="Derivatives.tanh">tanh</a></b>


<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mtable rowalign="center" ><mtr><mtd columnalign="left" >
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mo stretchy="false">[</mo>
<mi>tanh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>x</mi>
</mrow>
</mfrac>
</mtd><mtd columnalign="center" >
<mo stretchy="false">=</mo>
</mtd><mtd columnalign="right" >
<mn>1</mn>
<mo stretchy="false">-</mo>
<mi>tanh</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<msup><mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mtd></mtr></mtable>
</mrow></math>


<hr/>Input File: cppad/local/std_math_ad.hpp

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