document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=316781>Question 316781</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Will someone please show me how to find the derivative of the following function:<BR>\n" );
document.write( "f(x) = x^6 + x^4 + 1\r<BR>\n" );
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document.write( "A solution with full working and steps would be much appreciated.\r<BR>\n" );
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document.write( "Thanks -Nick. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #226731 by </B><A HREF=/tutors/aboutme.mpl?userid=moshiz08><B>moshiz08(60)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=moshiz08><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=moshiz08><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=226731\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Hi Nick, \r<BR>\n" );
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document.write( "This is a good question. I have to ask you what background you have in calculus. In particular, are you familiar with the power rule? Chain rule, product rule, derivative of exponential and logarithmic functions? \r<BR>\n" );
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document.write( "The power rule is very simple. The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5En ALIGN=MIDDLE  ALT=\"x%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Ax%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"n%2Ax%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6 ALIGN=MIDDLE  ALT=\"x%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E5 ALIGN=MIDDLE  ALT=\"6x%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4 ALIGN=MIDDLE  ALT=\"x%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E3 ALIGN=MIDDLE  ALT=\"4x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the same as the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E0 ALIGN=MIDDLE  ALT=\"x%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2Ax%5E%28-1%29 ALIGN=MIDDLE  ALT=\"0%2Ax%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So to answer your question, the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6+%2B+x%5E4+%2B+1 ALIGN=MIDDLE  ALT=\"x%5E6+%2B+x%5E4+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is simply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%5E5+%2B+4x%5E3+%2B+0 ALIGN=MIDDLE  ALT=\"6x%5E5+%2B+4x%5E3+%2B+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Now there are various ways to prove the power rule, and I would really like for you to see one. Merely memorizing a formula says nothing about our knowledge of mathematics. This is why I asked about your background, because if you have the right preparation already, the proof can be easy. If you do not have this preparation, it is not difficult to develop. \r<BR>\n" );
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document.write( "-Mo  </SPAN>\n" );
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