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 #266728 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=266728\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%5Ex ALIGN=MIDDLE  ALT=\"y=2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28y%29=ln%282%5Ex%29 ALIGN=MIDDLE  ALT=\"ln%28y%29=ln%282%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28y%29=x%2Aln%282%29 ALIGN=MIDDLE  ALT=\"ln%28y%29=x%2Aln%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pull down the exponent.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{y^{\prime}}{y}=\ln(2)\"> ... Derive both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=y\ln(2)\"> ... Solve for y'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=2^x\ln(2)\"> ... Now plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%5Ex ALIGN=MIDDLE  ALT=\"y=2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ex ALIGN=MIDDLE  ALT=\"b%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ex%2Aln%28b%29 ALIGN=MIDDLE  ALT=\"b%5Ex%2Aln%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "ii)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5Ex ALIGN=MIDDLE  ALT=\"y=x%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28y%29=ln%28x%5Ex%29 ALIGN=MIDDLE  ALT=\"ln%28y%29=ln%28x%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the natural log of both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28y%29=x%2Aln%28x%29 ALIGN=MIDDLE  ALT=\"ln%28y%29=x%2Aln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pull down the exponent.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{y^{\prime}}{y}=\ln(x)+1\"> ... Derive both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=y(\ln(x)+1)\"> ... Solve for y'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=x^x(\ln(x)+1)\"> ... Now plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5Ex ALIGN=MIDDLE  ALT=\"y=x%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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