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	<title>User:Osvaldo9022 - Revision history</title>
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	<updated>2026-06-21T11:32:16Z</updated>
	<subtitle>Revision history for this page on the wiki</subtitle>
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		<id>http://allanpatrick.net/index.php?title=User:Osvaldo9022&amp;diff=108815&amp;oldid=prev</id>
		<title>Osvaldo9022: Created page with &quot;For the function:&lt;br&gt;&lt;br&gt;y = x^x^x (the superscript notation on this text editor does not work with double superscripts)&lt;br&gt;&lt;br&gt;To solve for the derivative y&#039;, implicit differentiation is needed. First, the equation must be manipulated so there are no x&#039;s raised to x&#039;s on the right side of the equation. So, both sides of the equation must be input into a natural logarithm, wherein we can use the properties of logarithms to remove the superscripted powers of the right sid...&quot;</title>
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		<updated>2025-03-25T06:30:37Z</updated>

		<summary type="html">&lt;p&gt;Created page with &amp;quot;For the function:&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y = x^x^x (the superscript notation on this text editor does not work with double superscripts)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;To solve for the derivative y&amp;#039;, implicit differentiation is needed. First, the equation must be manipulated so there are no x&amp;#039;s raised to x&amp;#039;s on the right side of the equation. So, both sides of the equation must be input into a natural logarithm, wherein we can use the properties of logarithms to remove the superscripted powers of the right sid...&amp;quot;&lt;/p&gt;
&lt;p&gt;&lt;b&gt;New page&lt;/b&gt;&lt;/p&gt;&lt;div&gt;For the function:&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y = x^x^x (the superscript notation on this text editor does not work with double superscripts)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;To solve for the derivative y&amp;#039;, implicit differentiation is needed. First, the equation must be manipulated so there are no x&amp;#039;s raised to x&amp;#039;s on the right side of the equation. So, both sides of the equation must be input into a natural logarithm, wherein we can use the properties of logarithms to remove the superscripted powers of the right side:&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = ln(x^x^x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = xxln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y)/ln(x) = xx&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(ln(y)/ln(x)) = xln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;eln(ln(y)/ln(x)) = exln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y)/ln(x) = exln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = ln(x)exln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Now there are no functions raised to functions (x&amp;#039;s raised to x&amp;#039;s). Deriving this equation yields:&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;(1/y)(y&amp;#039;) = ln(x)exln(x)(x(1/x) + ln(x)) + exln(x)(1/x) = ln(x)exln(x)(1 + ln(x)) + exln(x)(1/x) = exln(x)(ln(x)(1+ln(x)) + (1/x))&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Solving for y&amp;#039; yields:&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y&amp;#039; = y[exln(x)(ln2(x) + ln(x) + (1/x))]&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;or&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y = xx^x&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = ln(x)x^x&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = xxln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;ln(y) = exlnxln(x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y&amp;#039;/y = exlnx[ln(x) + 1)ln(x) + exlnx(1/x)&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y&amp;#039; = y[exlnx(ln2(x) + ln(x) + 1/x)]&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;y&amp;#039; = xx^x[exlnx(ln2(x) + ln(x) + 1/x)]&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Also visit my web site: [https://km88.media/ live móc lồn]&lt;/div&gt;</summary>
		<author><name>Osvaldo9022</name></author>
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