document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=899462>Question 899462</A>: <I><SPAN STYLE=\"word-break: break-all;\"> f(t)= &#8730;t^2+2ln(t)\r<BR>\n" );
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document.write( "Find the derivative of each function.\r<BR>\n" );
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document.write( "Can someone help me figuring this equation out.  I would really appreciate the help.  Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #545423 by </B><A HREF=/tutors/aboutme.mpl?userid=swincher4391><B>swincher4391(1107)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=swincher4391><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=swincher4391><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=545423\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assuming the problem is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28t%5E2+%2B+2ln%28t%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28t%5E2+%2B+2ln%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we know that we are going to have to do the chain rule. I find it easiest to use simple substitution to do this. You'll see what I mean in a second.\r<BR>\n" );
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document.write( "Let's let u = t^2 + 2ln(t) so du = 2t + 2/t dt  [but dt is just 1 since derivative of t = 1]\r<BR>\n" );
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document.write( "Let's let v = sqrt(u) so dv = 1/2sqrt(u) * du\r<BR>\n" );
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document.write( "Going from dv we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv+=+1%2F2sqrt%28u%29+%2A+du ALIGN=MIDDLE  ALT=\"dv+=+1%2F2sqrt%28u%29+%2A+du\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substitute du = 2t + 2/t\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%282%2Asqrt%28u%29%29+%2A+%282t+%2B+2%2Ft%29 ALIGN=MIDDLE  ALT=\"1%2F%282%2Asqrt%28u%29%29+%2A+%282t+%2B+2%2Ft%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Substituting u in terms of t, we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%282%2Asqrt%28t%5E2%2B2ln%28t%29%29%29+%2A+%282t+%2B+2%2Ft%29 ALIGN=MIDDLE  ALT=\"1%2F%282%2Asqrt%28t%5E2%2B2ln%28t%29%29%29+%2A+%282t+%2B+2%2Ft%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Getting rid of 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%2B1%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29 ALIGN=MIDDLE  ALT=\"%28t%2B1%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <--- perfect acceptable\r<BR>\n" );
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document.write( "Getting the Wolfram-Alpha answer:\r<BR>\n" );
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document.write( "Get a common denominator in the numerator.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28t%5E2%2B1%29%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29 ALIGN=MIDDLE  ALT=\"%28%28t%5E2%2B1%29%2Ft%29%2F%28sqrt%28t%5E2%2B2ln%28t%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Drop the t into the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t%5E2%2B1%29%2F%28t%2Asqrt%28t%5E2%2B2ln%28t%29%29%29 ALIGN=MIDDLE  ALT=\"%28t%5E2%2B1%29%2F%28t%2Asqrt%28t%5E2%2B2ln%28t%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, to recap, any time you have functions within functions, we just assign them variables.\r<BR>\n" );
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document.write( "Like  sin(cos(x)))\r<BR>\n" );
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document.write( "We let u = cos(x) so du = -sin(x)\r<BR>\n" );
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document.write( "Now we let v = sin(u) so dv = cos(u) du\r<BR>\n" );
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document.write( "We go from dv and get cos(u) * (-sin(x))\r<BR>\n" );
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document.write( "Substitute u back in and get cos(cos(x)) * -sin(x).\r<BR>\n" );
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document.write( "Clean it up a bit and get -cos(cos(x))*sin(x).\r<BR>\n" );
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document.write( "Hope this makes sense and makes the chain rule a little easier! Feel free to email me if it does not.  <swincher4391@yahoo.com> </SPAN>\n" );
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