document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=79723>Question 79723</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Finding the derivative using the defintion?\r<BR>\n" );
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document.write( "hi i would like to find out how to solve this problem. i would like to find the derivative of the function using the definition of the derivative and state the domain of the function and the domain of its derivative\r<BR>\n" );
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document.write( "g(x) = 1/x^2\r<BR>\n" );
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document.write( "thanks for any help. it is very much appreciated  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #57192 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=57192\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using the definition of the derivative:\r<BR>\n" );
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document.write( "f'(x)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh%29 ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%28x%2Bh%29%5E2-1%2Fx%5E2%29%2Fh ALIGN=MIDDLE  ALT=\"%281%2F%28x%2Bh%29%5E2-1%2Fx%5E2%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with this expression by plugging in (x+h) into x of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2%2Fx%5E2%29%281%2F%28x%2Bh%29%5E2%29-%281%2Fx%5E2%29%28%28x%2Bh%29%5E2%2F%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both fractions to get a common denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2%2F%28x%5E2%28x%2Bh%29%5E2%29%29-%28%28x%2Bh%29%5E2%2F%28x%5E2%28x%2Bh%29%5E2%29%29%29%2Fh ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2-%28x%2Bh%29%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28%28x%5E2-%28x%2Bh%29%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the fractions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2-%28x%5E2%2B2xh%2Bh%5E2%29%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28%28x%5E2-%28x%5E2%2B2xh%2Bh%5E2%29%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bh%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bh%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E2-x%5E2-2xh-h%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28%28x%5E2-x%5E2-2xh-h%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the negative\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2xh-h%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28%28-2xh-h%5E2%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h%28-2x-h%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28h%28-2x-h%29%2F%28x%5E2%28x%2Bh%29%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out an h\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h%28-2x-h%29%2F%28h%28x%5E2%28x%2Bh%29%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"%28h%28-2x-h%29%2F%28h%28x%5E2%28x%2Bh%29%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the first fraction by the reciprocal of the 2nd\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%28h%29%28-2x-h%29%2F%28cross%28h%29%28x%5E2%28x%2Bh%29%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"%28cross%28h%29%28-2x-h%29%2F%28cross%28h%29%28x%5E2%28x%2Bh%29%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice these h terms cancel\r<BR>\n" );
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document.write( "So we're left with this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2x-h%29%2F%28%28x%5E2%28x%2Bh%29%5E2%29%29%29 ALIGN=MIDDLE  ALT=\"%28%28-2x-h%29%2F%28%28x%5E2%28x%2Bh%29%5E2%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2x-h%29%2F%28%28x%5E2%28x%5E2%2B2xh%2Bh%5E2%29%29%29%29 ALIGN=MIDDLE  ALT=\"%28%28-2x-h%29%2F%28%28x%5E2%28x%5E2%2B2xh%2Bh%5E2%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Foil the denominator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2x-h%29%2F%28%28x%5E4%2B2x%5E3h%2Bx%5E2h%5E2%29%29%29%29 ALIGN=MIDDLE  ALT=\"%28%28-2x-h%29%2F%28%28x%5E4%2B2x%5E3h%2Bx%5E2h%5E2%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-2x-%280%29%29%2F%28%28x%5E4%2B2x%5E3%280%29%2Bx%5E2%280%29%5E2%29%29%29%29 ALIGN=MIDDLE  ALT=\"%28%28-2x-%280%29%29%2F%28%28x%5E4%2B2x%5E3%280%29%2Bx%5E2%280%29%5E2%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now plug in h=0 since we're taking a limit to zero\r<BR>\n" );
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document.write( "So it all simplifies to this:\r<BR>\n" );
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document.write( "Now divide to simplify\r<BR>\n" );
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document.write( "So according to the definition, the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fx%5E2 ALIGN=MIDDLE  ALT=\"1%2Fx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
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