document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=726751>Question 726751</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a question regarding finding the second derivative of a hyperbola...\r<BR>\n" );
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document.write( "Find the value of d^2y/dx^2 for the hyperbola defined by the equation y^2-x^2=7 at the point (3,4).\r<BR>\n" );
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document.write( "The first derivative I have already found- dy/dx=x/y\r<BR>\n" );
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document.write( "However, to find the second derivative, I was a little bit more confused...<BR>\n" );
document.write( "y=xy^-1\r<BR>\n" );
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document.write( "d/dx(y)=d/dx(xy^-1)\r<BR>\n" );
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document.write( "dy/dx=x*dy/dx*-1y^-2 + y^-1 * d/dx (x)\r<BR>\n" );
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document.write( "dy/dx= -x*dy/dx*y^-2 +y^-1\r<BR>\n" );
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document.write( "dy/dx- (dy/dx (-xy^-2))= y^-1\r<BR>\n" );
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document.write( "dy/dx(1+xy^-2)= y^-1\r<BR>\n" );
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document.write( "dy/dx= y^-1/1+xy^-2<BR>\n" );
document.write( "dy/dx= 1/y(1+xy^-2)\r<BR>\n" );
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document.write( "Now, to plug in the points (3,4)\r<BR>\n" );
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document.write( "dy/dx= 1/y(1+xy^-2)\r<BR>\n" );
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document.write( "1/(4(1+(3)(4^-2))\r<BR>\n" );
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document.write( "= 1/(4+ (12/6)\r<BR>\n" );
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document.write( "I hope this was not too confusing- but I would like to know if this is the correct method for solving this problem, and if the answer is correct as well.\r<BR>\n" );
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document.write( "I would really appreciate your help.\r<BR>\n" );
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document.write( "Thank you very much.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #444791 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=444791\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the value of d^2y/dx^2 for the hyperbola defined by the equation y^2-x^2=7 at the point (3,4).<BR>\n" );
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document.write( "2ydy - 2xdx = 0<BR>\n" );
document.write( "y' = x/y (just checking)<BR>\n" );
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document.write( "y' = xy^-1<BR>\n" );
document.write( "y'' = y^-1*dx + x*(-1)*y^-2*dy\r<BR>\n" );
document.write( "\n" );
document.write( "The first derivative I have already found- dy/dx=x/y\r<BR>\n" );
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document.write( "However, to find the second derivative, I was a little bit more confused...<BR>\n" );
document.write( "y=xy^-1\r<BR>\n" );
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document.write( "d/dx(y)=d/dx(xy^-1)\r<BR>\n" );
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document.write( "dy/dx=x*dy/dx*-1y^-2 + y^-1 * d/dx (x)\r<BR>\n" );
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document.write( "dy/dx= -x*dy/dx*y^-2 +y^-1\r<BR>\n" );
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document.write( "dy/dx- (dy/dx (-xy^-2))= y^-1\r<BR>\n" );
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document.write( "dy/dx(1+xy^-2)= y^-1\r<BR>\n" );
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document.write( "dy/dx= y^-1/1+xy^-2<BR>\n" );
document.write( "dy/dx= 1/y(1+xy^-2)<BR>\n" );
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document.write( "(3/4) is in the 1st Q, so solve for the + half of the hyperbola<BR>\n" );
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document.write( "y = (x^2 + 7)^1/2<BR>\n" );
document.write( "y' = (1/2)*(x^2+7)^(-1/2)*2x<BR>\n" );
document.write( "y' = x*(x^2+7)^(-1/2)<BR>\n" );
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document.write( "y'' = (x^2+7)^(-1/2) + x*(-1/2)*(x^2+7)^(-3/2)*2x<BR>\n" );
document.write( "y'' = ((x^2+7)^(-1/2) - (x^2))/(x^2+7)^(3/2)<BR>\n" );
document.write( "y'' = (x^2+7 - x^2)/((x^2+7)^(3/2))<BR>\n" );
document.write( "y'' = 7/(x^2+7)^3/2)<BR>\n" );
document.write( "y''(3) = 7/(16)^3/2<BR>\n" );
document.write( "= 7/64 </SPAN>\n" );
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