document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=827885>Question 827885</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose y=x/u, where u is a function of x. Show that y(du/dx)+ u(dy/dx)=1 \r<BR>\n" );
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document.write( "Thanks!!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #498975 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=498975\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The easiest way to do this is to use implicit differentiation. (I will do with without implicit differentiation later.)<br><BR>\n" );
document.write( "If we multiply each side of the equation by u we get:<BR>\n" );
document.write( "y*u = x<BR>\n" );
document.write( "Implicitly differentiate. On the left side we will use the product rule:<BR>\n" );
document.write( "y*(du/dx) + u*(dy/dx) = 1<BR>\n" );
document.write( "And we're done!<br><BR>\n" );
document.write( "Using \"regular\" differentiation (the quotient rule) on y = x/u, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+%28u%281%29+-+x%28du%2Fdx%29%29%2Fu%5E2 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+%28u%281%29+-+x%28du%2Fdx%29%29%2Fu%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we use some algebra to transform this equation into the desired one. Since the desired equation has two terms on one side, I will split the fraction (i.e. \"un-subtract\"):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+u%2Fu%5E2+-+%28x%28du%2Fdx%29%29%2Fu%5E2 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+u%2Fu%5E2+-+%28x%28du%2Fdx%29%29%2Fu%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+1%2Fu+-+%28x%28du%2Fdx%29%29%2Fu%5E2 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+1%2Fu+-+%28x%28du%2Fdx%29%29%2Fu%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Factoring the second fraction:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+1%2Fu+-+%28x%2Fu%29%2A%281%2Fu%29%28du%2Fdx%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+1%2Fu+-+%28x%2Fu%29%2A%281%2Fu%29%28du%2Fdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since y = x/u we can substitute in for the x/u:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+1%2Fu+-+y%2A%281%2Fu%29%28du%2Fdx%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+1%2Fu+-+y%2A%281%2Fu%29%28du%2Fdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying both sides by u (to eliminate the fractions):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2A%28dy%2Fdx%29+=+1+-+y%2A%28du%2Fdx%29 ALIGN=MIDDLE  ALT=\"u%2A%28dy%2Fdx%29+=+1+-+y%2A%28du%2Fdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding y*(du/dx) to both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2A%28du%2Fdx%29+%2B+u%2A%28dy%2Fdx%29+=+1 ALIGN=MIDDLE  ALT=\"y%2A%28du%2Fdx%29+%2B+u%2A%28dy%2Fdx%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And we're done! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );