<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<a NAME="E1:30"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">p</font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font>;<br/>
<a NAME="E2:30"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:30"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">x</font> holds <br/>( <a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:30_1"/><i><font color="Green">E40</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<a NAME="E3:30_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:30_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 <b>by </b><i><a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:30_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E5:30_1"/><i><font color="Green">E46</font></i>: 
 <a href="taylor_1.html#K3" target="_self">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>;<br/>
<a NAME="E7:30_1"/><i><font color="Green">E49</font></i>: 
 <a href="sin_cos.html#K27">exp_R</a>  <a href="fdiff_1.html#R1">is_differentiable_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="sin_cos.html#T70">SIN_COS:70</a></i>;<br/>
<b>hence </b><a NAME="E8:30_1"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>;<br/>

<i><font color="Green">E50</font></i>:  <a href="fdiff_1.html#K1">diff</a> <a href="sin_cos.html#K27">exp_R</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> = 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" target="_self">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
;<br/>
 <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span>,<font color="Maroon">x</font> = 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1">diff</a> <a href="taylor_1.html#K3" target="_self">ln</a> ,<font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="txt" href="taylor_1.html#E1:30"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fdiff_1.html#T23">FDIFF_1:23</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a>, </i>
;<br/>
<b>hence </b> <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span>,<font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="ref" href="fdiff_2.html#T13">FDIFF_2:13</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <font color="Maroon">x</font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K4">to_power</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K4">to_power</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T33">POWER:33</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="power.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T32">POWER:32</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:30"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_2_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="sin_cos.html#K27">exp_R</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b> <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="limfunc1.html#K4">right_open_halfline</a> 0
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:30_2"/>
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0 holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:30_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 0
 ;<br/>

<a NAME="E2:30_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span>
 
<b>by </b><i>, , <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:30_2_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="limfunc1.html#D3">LIMFUNC1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:30_2_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>
<a NAME="E6:30_2_2"/><i><font color="Green">E56</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T39">POWER:39</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="partfun2.html#T6">PARTFUN2:6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K3" target="_self">ln</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><a href="power.html#K6">log</a> <a href="power.html#K8">number_e</a> ,<font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<a href="sin_cos.html#K27">exp_R</a>  <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="power.html#K5">log</a> <a href="power.html#K8">number_e</a> ,<span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T63">POWER:63</a></i>
<br/>.= 
<font color="Maroon">d</font> <a href="power.html#K3">to_power</a> <font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E1:30"><i><font color="Green">E31</font></i></a>, </i>
<br/>.= 
<font color="Maroon">d</font> <a href="prepower.html#K12">#R</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="power.html#D2">POWER:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_2"/>
<a href="sin_cos.html#K27">exp_R</a>  <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="seq_1.html#K13">(#)</a> <a href="taylor_1.html#K3" target="_self">ln</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies (  <a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K4" target="_self">#R</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="prepower.html#K12">#R</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>


</div>
