document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=375065>Question 375065</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello everyone, finding the following question puzzling, i have not covered this before, and am trying to get a heads up for when i start my course\r<BR>\n" );
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document.write( "Use logarithmic differentiation to differentiate the following:<BR>\n" );
document.write( "(i) y = 2^x (ii) y = x^x\r<BR>\n" );
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document.write( "Cheers </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #266735 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=266735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You get one question per post.  Since the first one is trivial, I'll do the second.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x^x\">\r<BR>\n" );
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document.write( "Take the natural log of both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln(y)\ =\ \ln\left(x^x\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln(y)\ =\ x\ln(x)\">\r<BR>\n" );
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document.write( "Differentiate both sides with respect to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> using the Chain Rule on the left and the Product Rule on the right:\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large u\ =\ \ln(y)\">.\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{du}{dy}\ =\ \frac{1}{y}\">\r<BR>\n" );
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document.write( "Hence:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dy}{dx}\,\cdot\,\left(\frac{1}{y}\right)\ =\ x\,\cdot\,\left(\frac{1}{x}\right)\ +\ \ln(x)\ =\ 1\ +\ \ln(x)\">\r<BR>\n" );
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document.write( "Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ x^x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{dy}{dx}\ =\ x^x\left(1\ +\ \ln(x)\right)\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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