document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=998820>Question 998820</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Hi, I need to find the first and second derivative of: f(x)=8x[4^(-x)]\r<BR>\n" );
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document.write( "and the answer in the book is:\r<BR>\n" );
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document.write( "f'(x)=-8[4^(-x)](x ln 4-1)\r<BR>\n" );
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document.write( "and the second one is:\r<BR>\n" );
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document.write( "f''(x)=8[4^(-x)] ln 4(x ln 4-2)\r<BR>\n" );
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document.write( "can someone show me the steps to get to those derivatives please?\r<BR>\n" );
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document.write( "Thank you!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #616580 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=616580\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = 8x[4^(-x)] = 8x / (4^x)<BR>\n" );
document.write( "note that the calculus division rule will be used, that is<BR>\n" );
document.write( "first derivative of (numerator(N) / denominator(D)) =<BR>\n" );
document.write( "(N' * D)-(N * D') / (D^2)<BR>\n" );
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document.write( "derivative of denominator calculation<BR>\n" );
document.write( "let y = 4^x<BR>\n" );
document.write( "take natural log of both sides of =<BR>\n" );
document.write( "ln y = x * ln(4)<BR>\n" );
document.write( "then use implicit differentiation<BR>\n" );
document.write( "(1/y) * y' = ln(4)<BR>\n" );
document.write( "y' = y * ln(4)<BR>\n" );
document.write( "note that y = 4^x, then<BR>\n" );
document.write( "y' = (4^x) * ln(4)<BR>\n" );
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document.write( "derivative of numerator calculation<BR>\n" );
document.write( "y = 8x<BR>\n" );
document.write( "y' = 8<BR>\n" );
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document.write( "f'(x) = (8 * (4^x)) - (8x * (4^x) * ln(4) ) / (4^2x)<BR>\n" );
document.write( "simplify this (see the following steps)<BR>\n" );
document.write( "f'(x) = (8 - 8x * ln(4)) / (4^x)<BR>\n" );
document.write( "here we have divided the numerator by (4^x) leaving (4^x) in the D<BR>\n" );
document.write( "f'(x) = (-8 * (x*ln(4)-1) / (4^x)<BR>\n" );
document.write( "factored -8, therefore<BR>\n" );
document.write( "f'(x) = -8 * [4^(-x)] * (x*ln(4)-1)<BR>\n" );
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document.write( "start with f'(x) =  (8 - 8x * ln(4)) / (4^x), then take first derivative of this using calculus division rule and implicit derivative of D<BR>\n" );
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document.write( "derivative of denominator calculation<BR>\n" );
document.write( "let y = 4^x<BR>\n" );
document.write( "y' = (4^x) * ln(4)<BR>\n" );
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document.write( "derivative of numerator calculation<BR>\n" );
document.write( "y = 8 - 8x * ln(4)<BR>\n" );
document.write( "note that derivative of a constant is 0 <BR>\n" );
document.write( "y' = 0 -8*ln(4)<BR>\n" );
document.write( "y' = -8*ln(4)<BR>\n" );
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document.write( "f''(x) = (-(8*ln(4))*(4^x)) - ((8 - 8x * ln(4))*(4^x) * ln(4)) / (4^2x)<BR>\n" );
document.write( "f''(x) = -8*ln(4)*4^x -8*ln(4)*4^x +8x*ln^2(4)*4^x / 4^2x<BR>\n" );
document.write( "f''(x) = -8*ln(4) -8*ln(4) + 8x*ln^2(4) / 4^x<BR>\n" );
document.write( "f''(x) = 8x*ln^2(4) -16*ln(4) / 4^x<BR>\n" );
document.write( "f''(x) = 8 * [4^(-x)] * ln(4) * (x*ln(4) - 2) \r<BR>\n" );
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