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<center><b><big><big>First Order Partial Derivative: Driver Routine</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'>dy</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.ForOne(</span></font><i><font color="black"><span style='white-space: nowrap'>x</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>j</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/>
We use 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
<mo stretchy="false">:</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>n</mi>
</msup>
<mo stretchy="false">&#x02192;</mo>
<msup><mi mathvariant='italic'>B</mi>
<mi mathvariant='italic'>m</mi>
</msup>
</mrow></math>

 to denote the
<a href="glossary.xml#AD Function" target="_top"><span style='white-space: nowrap'>AD&#xA0;function</span></a>
 corresponding to 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
The syntax above sets 
<code><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i></code>
 to the
partial of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
</mrow></math>

 with respect to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>

; i.e.,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>dy</mi>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<mi mathvariant='italic'>F</mi>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">=</mo>
<mrow><mo stretchy="true">[</mo><mrow><mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>F</mi>
<mn>0</mn>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x022EF;</mo>
<mo stretchy="false">,</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>F</mi>
<mrow><mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow><mo stretchy="true">]</mo></mrow>
</mrow></math>

<br/>
<b><big><a name="f" id="f">f</a></big></b>
<br/>
The object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;ADFun&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'>f</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
Note that the <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
 object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 is not <code><font color="blue">const</font></code>
(see <a href="forone.xml#ForOne Uses Forward" target="_top"><span style='white-space: nowrap'>ForOne&#xA0;Uses&#xA0;Forward</span></a>
 below).

<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 prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#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>
(see <a href="forone.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below)
and its size 
must be equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
, the dimension of the
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
It specifies
that point at which to evaluate the partial derivative.

<br/>
<br/>
<b><big><a name="j" id="j">j</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>j</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
an is less than 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
,
<a href="seq_property.xml#Domain" target="_top"><span style='white-space: nowrap'>domain</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
It specifies the component of 
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
 
for which we are computing the partial derivative.

<br/>
<br/>
<b><big><a name="dy" id="dy">dy</a></big></b>
<br/>
The result 
<code><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see <a href="forone.xml#Vector" target="_top"><span style='white-space: nowrap'>Vector</span></a>
 below)
and its size is 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
</mrow></math>

, the dimension of the
<a href="seq_property.xml#Range" target="_top"><span style='white-space: nowrap'>range</span></a>
 space for 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
.
The value of 
<code><i><font color="black"><span style='white-space: nowrap'>dy</span></font></i></code>
 is the partial of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>F</mi>
</mrow></math>

 with respect to

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow></math>

 evaluated at 
<code><i><font color="black"><span style='white-space: nowrap'>x</span></font></i></code>
; i.e.,
for 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">&#x02026;</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">-</mo>
<mn>1</mn>
</mrow></math>



<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mo stretchy="false">.</mo>
<mi mathvariant='italic'>dy</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">]</mo>
<mo stretchy="false">=</mo>
<mfrac><mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>F</mi>
<mi mathvariant='italic'>i</mi>
</msub>
</mrow>
<mrow><mo stretchy="false">&#x02202;</mo>
<msub><mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>j</mi>
</msub>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>x</mi>
<mo stretchy="false">)</mo>
</mrow></math>

<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
 must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
 class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements&#xA0;of&#xA0;type</span></a>


<code><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i></code>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
 will generate an error message
if this is not the case.

<br/>
<br/>
<b><big><a name="ForOne Uses Forward" id="ForOne Uses Forward">ForOne Uses Forward</a></big></b>
<br/>
After each call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
,
the object 
<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i></code>
 contains the corresponding 
<a href="glossary.xml#Taylor Coefficient" target="_top"><span style='white-space: nowrap'>Taylor&#xA0;coefficients</span></a>
.
After a call to <code><font color="blue">ForOne</font></code>,
the zero order Taylor coefficients correspond to

<code><i><font color="black"><span style='white-space: nowrap'>f</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(0,</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>

and the other coefficients are unspecified.

<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>

<br/>
The routine 
<a href="for_one.cpp.xml" target="_top"><span style='white-space: nowrap'>ForOne</span></a>
 is both an example and test.
It returns <code><font color="blue">true</font></code>, if it succeeds and <code><font color="blue">false</font></code> otherwise.


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

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