document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1113571>Question 1113571</A>: <I><SPAN STYLE=\"word-break: break-all;\"> CALCULUS. Find (d^2 y)/(dx^2) if y=(x^2 +1)^5. Please help. Thankyou in advance! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #728637 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=728637\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%28x%5E2%2B1%29%5E5+ ALIGN=MIDDLE  ALT=\"+y+=+%28x%5E2%2B1%29%5E5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "—<BR>\n" );
document.write( "For the first part we can use u-substitution:<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+u+=+%28x%5E2%2B1%29+ ALIGN=MIDDLE  ALT=\"+u+=+%28x%5E2%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "     <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+du+=+2x+dx+ ALIGN=MIDDLE  ALT=\"+du+=+2x+dx+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+u%5E5+ ALIGN=MIDDLE  ALT=\"+y+=+u%5E5+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++dy+=+5%28u%5E4%29+du++=+5%28x%5E2%2B1%29%5E4%282x+dx%29+=+10x%28%28x%5E2%2B1%29%5E4%29+dx ALIGN=MIDDLE  ALT=\"++dy+=+5%28u%5E4%29+du++=+5%28x%5E2%2B1%29%5E4%282x+dx%29+=+10x%28%28x%5E2%2B1%29%5E4%29+dx\"   DISABLED_style= \"max-width:90%; height:auto;\" > —> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+dy%2Fdx+=+10x%28x%5E2%2B1%29%5E4+ ALIGN=MIDDLE  ALT=\"+dy%2Fdx+=+10x%28x%5E2%2B1%29%5E4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "Aside:<BR>\n" );
document.write( "Notice that the chain rule (which applies to composite functions) could have been used:<BR>\n" );
document.write( "The chain rule says (f(g(x)))' = f'(g)*g')  \"The derivative of f of g(x) is the derivative of f(g(x)) times the derivative of g(x)\"   Often used for expressions like (mx+k)^n where, m and k are numbers, g(x)=mx+k and f(x)=x^n,  one takes  n*(mx+k)^(n-1) then multiplies by dg/dx = d(mx+k)/dx = m to arrive at  m(mx+k)^(n-1).   Think of it as an implicit use of u-substitution.  <BR>\n" );
document.write( "—<BR>\n" );
document.write( "I mention the chain rule because I will use it for finding the 2nd derivative.  However, that step happens to be identical in procedure to how we found the first derivative.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Now onward to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2y%2Fdx%5E2+ ALIGN=MIDDLE  ALT=\"d%5E2y%2Fdx%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > …<BR>\n" );
document.write( "—<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+dy%2Fdx+=+10x%28x%5E2%2B1%29%5E4+ ALIGN=MIDDLE  ALT=\"+dy%2Fdx+=+10x%28x%5E2%2B1%29%5E4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  is a product of two functions 10x and (x^2+1)^4<BR>\n" );
document.write( "—<BR>\n" );
document.write( "The product of functions lends itself to using the product rule:<BR>\n" );
document.write( "The product rule is  (f*g)' = fg' + f'g   (where f=f(x) and f' = df/dx, g=g(x), g'=dg/dx) <BR>\n" );
document.write( "\"The first times the derivative of the 2nd, plus the 2nd times the derivative of the first.\"<BR>\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "The first is \"10x\" , the 2nd is \"(x^2+1)^4\"  so we will do  10x * derivative of(x^2+1)^4) + (x^2+1)^4 * derivative of(10x),   noting that finding the derivative of (x^2+1)^4 wrt x is very similar to how we found the first derivative (it is the same process just different numbers).  The part I'm referring to is highlighted in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+green%28green%29+ ALIGN=MIDDLE  ALT=\"+green%28green%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > below.\r<BR>\n" );
document.write( "\n" );
document.write( "—\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2y%2Fdx%5E2+=+10x+%28green%284%2A%28x%5E2%2B1%29%5E3%2A%282x%29%29%29+%2B+%28x%5E2%2B1%29%5E4+%2A+10+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2y%2Fdx%5E2+=+%2880x%5E2%29%28x%5E2%2B1%29%5E3+%2B+10%28x%5E2%2B1%29%5E4+ ALIGN=MIDDLE  ALT=\"+d%5E2y%2Fdx%5E2+=+%2880x%5E2%29%28x%5E2%2B1%29%5E3+%2B+10%28x%5E2%2B1%29%5E4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <BR>\n" );
document.write( "Which can be re-written after much algebra:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2y%2Fdx%5E2+=+highlight%2890x%5E8+%2B+280x%5E6+%2B+300x%5E4+%2B+120x%5E2+%2B+10%29+ ALIGN=MIDDLE  ALT=\"+d%5E2y%2Fdx%5E2+=+highlight%2890x%5E8+%2B+280x%5E6+%2B+300x%5E4+%2B+120x%5E2+%2B+10%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "and that factors (thanks to an online factoring tool) to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+d%5E2y%2Fdx%5E2+=+highlight%2810%28x%5E2%2B1%29%5E3%289x%5E2%2B1%29%29+ ALIGN=MIDDLE  ALT=\"+d%5E2y%2Fdx%5E2+=+highlight%2810%28x%5E2%2B1%29%5E3%289x%5E2%2B1%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );