document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208298>Question 1208298</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if (x/y) + (y/x) = 1 , then y'' = (y/x) (True or False) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846630 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846630\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red>False</font><BR>\n" );
document.write( "If you said either <font color=red>y' = y/x</font> or <font color=red>y'' = 0</font>, then you would making a true statement.\r<BR>\n" );
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document.write( "Work Shown\r<BR>\n" );
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document.write( "w = x/y<BR>\n" );
document.write( "w' = (y-x*y')/(y^2) .... quotient rule\r<BR>\n" );
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document.write( "z = y/x<BR>\n" );
document.write( "z' = (y'*x-y)/(x^2) ... quotient rule\r<BR>\n" );
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document.write( "(x/y) + (y/x) = 1<BR>\n" );
document.write( "w + z = 1<BR>\n" );
document.write( "w' + z' = 0  ........ applied implicit derivative<BR>\n" );
document.write( "(y-x*y')/(y^2) + (y'*x-y)/(x^2) = 0<BR>\n" );
document.write( "( x^2(y-x*y') + y^2(y'*x-y) )/(x^2y^2) = 0<BR>\n" );
document.write( "x^2(y-x*y') + y^2(y'*x-y) = 0<BR>\n" );
document.write( "x^2*y-x^3*y' + xy^2*y'-y^3 = 0<BR>\n" );
document.write( "-x^3*y'+xy^2*y' = -x^2*y+y^3<BR>\n" );
document.write( "y'( -x^3+xy^2 ) = -x^2*y+y^3<BR>\n" );
document.write( "y' = (-x^2*y+y^3)/(-x^3+xy^2)<BR>\n" );
document.write( "y' = (-y(x^2-y^2))/(-x(x^2-y^2))<BR>\n" );
document.write( "<font color=red>y' = y/x</font>\r<BR>\n" );
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document.write( "Confirmation using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=derivative+%28x%2Fy%29+%2B+%28y%2Fx%29+%3D+1\">https://www.wolframalpha.com/input?i=derivative+%28x%2Fy%29+%2B+%28y%2Fx%29+%3D+1</a>\r<BR>\n" );
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document.write( "Another way to confirm is to use GeoGebra's ImplicitDerivative command.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://geogebra.github.io/docs/manual/en/commands/ImplicitDerivative/\">https://geogebra.github.io/docs/manual/en/commands/ImplicitDerivative/</a><BR>\n" );
document.write( "You would type in <font color=blue>ImplicitDerivative(x/y+y/x-1)</font> which produces the output y/x.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "y' = y/x<BR>\n" );
document.write( "y'' = (y'*x-y)/(x^2) .... quotient rule<BR>\n" );
document.write( "y'' = ((y/x)*x-y)/(x^2) ....... substitute in y' = y/x<BR>\n" );
document.write( "y'' = (y-y)/(x^2)<BR>\n" );
document.write( "y'' = (0)/(x^2)<BR>\n" );
document.write( "<font color=red>y'' = 0</font>\r<BR>\n" );
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document.write( "Another Approach\r<BR>\n" );
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document.write( "(x/y) + (y/x) = 1<BR>\n" );
document.write( "(x^2)/(xy)+(y^2)/(xy) = 1<BR>\n" );
document.write( "(x^2+y^2)/(xy) = 1\r<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "p = x^2+y^2<BR>\n" );
document.write( "q = xy<BR>\n" );
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document.write( "p' = 2x+2y*y'<BR>\n" );
document.write( "q' = y+x*y'\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "(x^2+y^2)/(xy) = 1<BR>\n" );
document.write( "d/dx[ (x^2+y^2)/(xy) ] = d/dx[1]<BR>\n" );
document.write( "d/dx[ p/q ] = 0<BR>\n" );
document.write( "(p'*q - p*q')/(q^2) = 0 ............. quotient rule<BR>\n" );
document.write( "p'*q - p*q' = 0<BR>\n" );
document.write( "(2x+2y*y')xy - (x^2+y^2)(y+x*y') = 0<BR>\n" );
document.write( "2x^2y+2xy^2*y' - (x^2(y+x*y')+y^2(y+x*y')) = 0<BR>\n" );
document.write( "2x^2y+2xy^2*y' - (x^2y+x^3*y')-(y^3+xy^2*y') = 0<BR>\n" );
document.write( "2x^2y+2xy^2*y' - x^2y-x^3*y'-y^3-xy^2*y' = 0<BR>\n" );
document.write( "2xy^2*y'-x^3*y'-xy^2*y' = -2x^2y+x^2y+y^3<BR>\n" );
document.write( "y'(2xy^2-x^3-xy^2) = -2x^2y+x^2y+y^3<BR>\n" );
document.write( "y'(xy^2-x^3) = -x^2y+y^3<BR>\n" );
document.write( "y' = (-x^2y+y^3)/(xy^2-x^3)<BR>\n" );
document.write( "y' = (-y(x^2-y^2))/(-x(-y^2+x^2))<BR>\n" );
document.write( "y' = (-y(x^2-y^2))/(-x(x^2-y^2))<BR>\n" );
document.write( "y' = y/x\r<BR>\n" );
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document.write( "From here the steps are the same as the previous section.<BR>\n" );
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