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<head>
<title>Section1.11</title>
<link rel="stylesheet" type="text/css" href="graphicstyle.css" />
<script type="text/javascript" src="bookax1.js" />
</head>
<body>
<a href="book-contents.xhtml" style="margin-right: 10px;">Book Contents</a><a href="section-1.10.xhtml" style="margin-right: 10px;">Previous Section 1.10 Series</a><a href="section-1.12.xhtml" style="margin-right: 10px;">Next Section 1.12 Integration</a>
<a href="book-index.xhtml">Book Index</a><div class="section" id="sec-1.11">
<h2 class="sectiontitle">1.11 Derivatives</h2>
<a name="ugIntroCalcDeriv" class="label"/>
<p>Use the Axiom function <span style="font-weight: bold;"> D</span> to differentiate an
<span class="index">derivative</span><a name="chapter-1-74"/> expression. <span class="index">differentiation</span><a name="chapter-1-75"/>
</p>
<p>To find the derivative of an expression <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math> with respect to a
variable <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, enter <span style="font-weight: bold;"> D</span>(f, x).
</p>
<div id="spadComm1-150" class="spadComm" >
<form id="formComm1-150" action="javascript:makeRequest('1-150');" >
<input id="comm1-150" type="text" class="command" style="width: 10em;" value="f := exp exp x" />
</form>
<span id="commSav1-150" class="commSav" >f := exp exp x</span>
<div id="mathAns1-150" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><msup><mi>e</mi><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<div id="spadComm1-151" class="spadComm" >
<form id="formComm1-151" action="javascript:makeRequest('1-151');" >
<input id="comm1-151" type="text" class="command" style="width: 5em;" value="D(f, x)" />
</form>
<span id="commSav1-151" class="commSav" >D(f, x)</span>
<div id="mathAns1-151" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mo></mo><mrow><msup><mi>e</mi><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>An optional third argument <math xmlns="&mathml;" mathsize="big"><mstyle><mi>n</mi></mstyle></math> in <span style="font-weight: bold;"> D</span> asks Axiom for the <math xmlns="&mathml;" mathsize="big"><mstyle><mi>n</mi></mstyle></math>-th
derivative of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math>. This finds the fourth derivative of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>f</mi></mstyle></math> with
respect to <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>.
</p>
<div id="spadComm1-152" class="spadComm" >
<form id="formComm1-152" action="javascript:makeRequest('1-152');" >
<input id="comm1-152" type="text" class="command" style="width: 7em;" value="D(f, x, 4)" />
</form>
<span id="commSav1-152" class="commSav" >D(f, x, 4)</span>
<div id="mathAns1-152" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mo>(</mo><mrow><msup><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mn>4</mn></msup></mrow><mo>+</mo><mrow><mn>6</mn><mo></mo><mrow><msup><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mn>3</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>7</mn><mo></mo><mrow><msup><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow><mo>)</mo></mrow><mo></mo><mrow><msup><mi>e</mi><mrow><msup><mi>e</mi><mi>x</mi></msup></mrow></msup></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>You can also compute partial derivatives by specifying the order of
<span class="index">differentiation:partial</span><a name="chapter-1-76"/>
differentiation.
</p>
<div id="spadComm1-153" class="spadComm" >
<form id="formComm1-153" action="javascript:makeRequest('1-153');" >
<input id="comm1-153" type="text" class="command" style="width: 12em;" value="g := sin(x**2 + y)" />
</form>
<span id="commSav1-153" class="commSav" >g := sin(x**2 + y)</span>
<div id="mathAns1-153" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mo>sin</mo><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<div id="spadComm1-154" class="spadComm" >
<form id="formComm1-154" action="javascript:makeRequest('1-154');" >
<input id="comm1-154" type="text" class="command" style="width: 5em;" value="D(g, y)" />
</form>
<span id="commSav1-154" class="commSav" >D(g, y)</span>
<div id="mathAns1-154" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mo>cos</mo><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<div id="spadComm1-155" class="spadComm" >
<form id="formComm1-155" action="javascript:makeRequest('1-155');" >
<input id="comm1-155" type="text" class="command" style="width: 12em;" value="D(g, [y, y, x, x])" />
</form>
<span id="commSav1-155" class="commSav" >D(g, [y, y, x, x])</span>
<div id="mathAns1-155" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mn>4</mn><mo></mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo></mo><mrow><mo>sin</mo><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>-</mo><mrow><mn>2</mn><mo></mo><mrow><mo>cos</mo><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>Axiom can manipulate the derivatives (partial and iterated) of
<span class="index">differentiation:formal</span><a name="chapter-1-77"/> expressions involving formal operators.
All the dependencies must be explicit.
</p>
<p>This returns <math xmlns="&mathml;" mathsize="big"><mstyle><mn>0</mn></mstyle></math> since F (so far) does not explicitly depend on <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>.
</p>
<div id="spadComm1-156" class="spadComm" >
<form id="formComm1-156" action="javascript:makeRequest('1-156');" >
<input id="comm1-156" type="text" class="command" style="width: 4em;" value="D(F,x)" />
</form>
<span id="commSav1-156" class="commSav" >D(F,x)</span>
<div id="mathAns1-156" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mn>0</mn></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Polynomial Integer
</div>
<p>Suppose that we have F a function of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math>, and <math xmlns="&mathml;" mathsize="big"><mstyle><mi>z</mi></mstyle></math>,
where <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math> and <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math> are themselves functions of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>z</mi></mstyle></math>.
</p>
<p>Start by declaring that <math xmlns="&mathml;" mathsize="big"><mstyle><mi>F</mi></mstyle></math>, <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, and <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math> are operators.
<span class="index">operator</span><a name="chapter-1-78"/>
</p>
<div id="spadComm1-157" class="spadComm" >
<form id="formComm1-157" action="javascript:makeRequest('1-157');" >
<input id="comm1-157" type="text" class="command" style="width: 35em;" value="F := operator 'F; x := operator 'x; y := operator 'y" />
</form>
<span id="commSav1-157" class="commSav" >F := operator 'F; x := operator 'x; y := operator 'y</span>
<div id="mathAns1-157" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mi>y</mi></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: BasicOperator
</div>
<p>You can use F, <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, and <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math> in expressions.
</p>
<div id="spadComm1-158" class="spadComm" >
<form id="formComm1-158" action="javascript:makeRequest('1-158');" >
<input id="comm1-158" type="text" class="command" style="width: 22em;" value="a := F(x z, y z, z**2) + x y(z+1)" />
</form>
<span id="commSav1-158" class="commSav" >a := F(x z, y z, z**2) + x y(z+1)</span>
<div id="mathAns1-158" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mi>x</mi><mo>(</mo><mrow><mi>y</mi><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mi>F</mi><mo>(</mo><mrow><mrow><mi>x</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>Differentiate formally with respect to <math xmlns="&mathml;" mathsize="big"><mstyle><mi>z</mi></mstyle></math>.
The formal derivatives appearing in <math xmlns="&mathml;" mathsize="big"><mstyle><mi>dadz</mi></mstyle></math> are not just formal symbols,
but do represent the derivatives of <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math>, and F.
</p>
<div id="spadComm1-159" class="spadComm" >
<form id="formComm1-159" action="javascript:makeRequest('1-159');" >
<input id="comm1-159" type="text" class="command" style="width: 10em;" value="dadz := D(a, z)" />
</form>
<span id="commSav1-159" class="commSav" >dadz := D(a, z)</span>
<div id="mathAns1-159" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mtable><mtr><mtd><mrow><mn>2</mn><mo></mo><mi>z</mi><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>3</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><mrow><msubsup><mi>x</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>1</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><mi>x</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mrow><mrow><msubsup><mi>x</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mrow><mi>y</mi><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo></mo><mrow><mrow><msubsup><mi>y</mi><mrow><mo></mo></mrow><mrow><mo>′</mo></mrow></msubsup></mrow><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>You can evaluate the above for particular functional values of
F, <math xmlns="&mathml;" mathsize="big"><mstyle><mi>x</mi></mstyle></math>, and <math xmlns="&mathml;" mathsize="big"><mstyle><mi>y</mi></mstyle></math>. If <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>x</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mstyle></math> is <span style="font-weight: bold;"> exp</span>(z) and <math xmlns="&mathml;" mathsize="big"><mstyle><mrow><mi>y</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mstyle></math> is <span style="font-weight: bold;"> log</span>(z+1),
then evaluates <span class="teletype">dadz</span>.
</p>
<div id="spadComm1-160" class="spadComm" >
<form id="formComm1-160" action="javascript:makeRequest('1-160');" >
<input id="comm1-160" type="text" class="command" style="width: 36em;" value="eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))" />
</form>
<span id="commSav1-160" class="commSav" >eval(eval(dadz, 'x, z +-> exp z), 'y, z +-> log(z+1))</span>
<div id="mathAns1-160" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>3</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo></mo><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>1</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<p>You obtain the same result by first evaluating <math xmlns="&mathml;" mathsize="big"><mstyle><mi>a</mi></mstyle></math> and
then differentiating.
</p>
<div id="spadComm1-161" class="spadComm" >
<form id="formComm1-161" action="javascript:makeRequest('1-161');" >
<input id="comm1-161" type="text" class="command" style="width: 34em;" value="eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))" />
</form>
<span id="commSav1-161" class="commSav" >eval(eval(a, 'x, z +-> exp z), 'y, z +-> log(z+1))</span>
<div id="mathAns1-161" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mrow><mrow><mi>F</mi><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo>+</mo><mi>z</mi><mo>+</mo><mn>2</mn></mrow></mstyle></math>
</td></tr>
</table>
</div>
<div class="returnType">
Type: Expression Integer
</div>
<div id="spadComm1-162" class="spadComm" >
<form id="formComm1-162" action="javascript:makeRequest('1-162');" >
<input id="comm1-162" type="text" class="command" style="width: 6em;" value="D(%, z)" />
</form>
<span id="commSav1-162" class="commSav" >D(%, z)</span>
<div id="mathAns1-162" ></div>
</div>
<div class="math">
<table>
<tr><td>
<math xmlns="&mathml;" mathsize="big" display="block"><mstyle><mfrac><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo></mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>+</mo><mrow><mn>2</mn><mo></mo><mi>z</mi></mrow><mo>)</mo></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>3</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>2</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow><mo>+</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mrow><mrow><mo>(</mo><mi>z</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo></mo><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo></mo><mrow><mrow><msub><mi>F</mi><mrow><mo>,</mo><mn>1</mn></mrow></msub></mrow><mo>(</mo><mrow><mrow><msup><mi>e</mi><mi>z</mi></msup></mrow><mo>,</mo><mrow><mo>log</mo><mo>(</mo><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>,</mo><mrow><msup><mi>z</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mi>z</mi><mo>+</mo><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mi>z</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></math>
</td></tr>
</table>
</div>
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Type: Expression Integer
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