document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=223602>Question 223602</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find dy/dx given that y=1+x^(2)e^(y) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #167138 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=167138\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find dy/dx we will need to understand:<ul><li>Derivatives of a sum of functions</li><li>Derivatives of a product of functions</li><li>The Chain Rule</li><li>Implicit differentiation</li></ul><BR>\n" );
document.write( "I hope you understand most of these because I am not going to go into great detail.<br><BR>\n" );
document.write( "We'll start by<ul><li>Presuming that y is a function of x</li><li>Breaking the rest of the equation into \"bite-size\" functions of x. (By \"bite-size\" I mean functions for which we can easily find the derivative.)</li></ul><BR>\n" );
document.write( "So let's define u(x) = 1 and v(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2e%5Ey ALIGN=MIDDLE  ALT=\"x%5E2e%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This allows us to rewrite the equation as:<BR>\n" );
document.write( "y = u + v<BR>\n" );
document.write( "And the derivative is easy for sums of functions:<BR>\n" );
document.write( "Equation y' = u' + v'<BR>\n" );
document.write( "u' is easy but v' is not. So we will break v into smaller pieces. Let's define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+x%5E2 ALIGN=MIDDLE  ALT=\"p%28x%29+=+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%28x%29+=+e%5Ey ALIGN=MIDDLE  ALT=\"q%28x%29+=+e%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes v = p*q. And the derivative of this product of functions:<BR>\n" );
document.write( "v' = p*q' + q*p'<BR>\n" );
document.write( "By the Chain rule,<BR>\n" );
document.write( "q' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ey%2Ay%27 ALIGN=MIDDLE  ALT=\"e%5Ey%2Ay%27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and p' is an easy derivative:<BR>\n" );
document.write( "p' = 2x<BR>\n" );
document.write( "so by substituting these into v' = p*q' + q*p' we get:<BR>\n" );
document.write( "v' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ae%5Ey ALIGN=MIDDLE  ALT=\"x%5E2%2Ae%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >*y' + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ey%2A2x ALIGN=MIDDLE  ALT=\"e%5Ey%2A2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now by substituting u' (which is 0) and v' (above) into y' = u' + v' we get:<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%2B+x%5E2%2Ae%5Ey ALIGN=MIDDLE  ALT=\"0+%2B+x%5E2%2Ae%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >*y' + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ey%2A2x ALIGN=MIDDLE  ALT=\"e%5Ey%2A2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ae%5Ey ALIGN=MIDDLE  ALT=\"x%5E2%2Ae%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >*y' + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xe%5Ey ALIGN=MIDDLE  ALT=\"2xe%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now all we need to do is solve for y'. Start by subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ae%5Ey ALIGN=MIDDLE  ALT=\"x%5E2%2Ae%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >*y' from both sides. This will gather the y' terms on the left side:<BR>\n" );
document.write( "y' - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ae%5Ey ALIGN=MIDDLE  ALT=\"x%5E2%2Ae%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" >*y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xe%5Ey ALIGN=MIDDLE  ALT=\"2xe%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we'll factor out y' on the left side:<BR>\n" );
document.write( "y'*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+-+x%5E2%2Ae%5Ey%29 ALIGN=MIDDLE  ALT=\"%281+-+x%5E2%2Ae%5Ey%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xe%5Ey ALIGN=MIDDLE  ALT=\"2xe%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And finally divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281+-+x%5E2%2Ae%5Ey%29 ALIGN=MIDDLE  ALT=\"%281+-+x%5E2%2Ae%5Ey%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "y' = dy/dx = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282xe%5Ey%29%2F%281+-+x%5E2%2Ae%5Ey%29 ALIGN=MIDDLE  ALT=\"%282xe%5Ey%29%2F%281+-+x%5E2%2Ae%5Ey%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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