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<b>thus </b><a NAME="E1:22"/>
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a>, <a class="ref" href="fdiff_2.html#T45" title="FDIFF_2:th.45">FDIFF_2:45</a></i>;<br/>

<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span>
 ;<br/>

<a NAME="E3:22"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:22"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
<a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a></i>
<br/>.= 
<font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E3:22"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E5:22"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K27" title="SIN_COS:func.27">exp_R</a>  <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a> </span>)</span>,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:22"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="taylor_1.html#T16" target="_self" title="TAYLOR_1:th.16">Th16</a>, <a class="ref" href="fdiff_2.html#T45" title="FDIFF_2:th.45">FDIFF_2:45</a></i>;<br/>


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