document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010200>Question 1010200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Need help with one more please:<BR>\n" );
document.write( "Differentiate as many times as necessary to evaluate f'''(-3) if f(x) = 2 ln( absolute value of x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #625664 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=625664\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x) = 2 * ln(|x|)<BR>\n" );
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document.write( "Calculate the first derivative of f(x)<BR>\n" );
document.write( "d/dx(f(x)) = d/dx(2 log(abs(x)))<BR>\n" );
document.write( "The derivative of f(x) is f'(x):<BR>\n" );
document.write( "f'(x) = d/dx(2 log(abs(x)))<BR>\n" );
document.write( "Factor out constants:<BR>\n" );
document.write( "f'(x) = 2 d/dx(log(abs(x)))<BR>\n" );
document.write( "Using the chain rule, d/dx(log(abs(x))) = ( dlog(u))/( du) ( du)/( dx), where u = abs(x) and ( d)/( du)(log(u)) = 1/u:<BR>\n" );
document.write( "f'(x) = 2 (d/dx(abs(x)))/(abs(x))<BR>\n" );
document.write( "The derivative of abs(x) is x/(abs(x)):<BR>\n" );
document.write( "f'(x) = (2 x/(abs(x)))/(abs(x))<BR>\n" );
document.write( "Simplify the expression:<BR>\n" );
document.write( "f'(x) = (2 x)/abs(x)^2<BR>\n" );
document.write( "Simplifying powers, abs(x)^2 = x^2:<BR>\n" );
document.write( "f'(x) = 2/x  <BR>\n" );
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document.write( "Calculate the second derivative of f(x)<BR>\n" );
document.write( "f'(x) = 2/x<BR>\n" );
document.write( "d/dx(f'(x)) = d/dx(2/x)<BR>\n" );
document.write( "The second derivative of f'(x) is f''(x):<BR>\n" );
document.write( "f''(x) = d/dx(2/x)<BR>\n" );
document.write( "Factor out constants:<BR>\n" );
document.write( "f''(x) = 2 d/dx(1/x)<BR>\n" );
document.write( "Use the power rule, d/dx(x^n) = n x^(n-1), where n = -1: d/dx(1/x) = d/dx(x^(-1)) = -x^(-2):<BR>\n" );
document.write( "f''(x) = 2 (-1)/(x^2)<BR>\n" );
document.write( "Expand the left hand side:<BR>\n" );
document.write( "f''(x) = -2/x^2<BR>\n" );
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document.write( "Calculate the third derivative of f(x)<BR>\n" );
document.write( "f''(x) = -2/x^2<BR>\n" );
document.write( "The third derivative of f(x) is f'''(x):<BR>\n" );
document.write( "f'''(x) = d/dx(-2/x^2)<BR>\n" );
document.write( "Factor out constants:<BR>\n" );
document.write( "f'''(x) = -2 d/dx(1/x^2)<BR>\n" );
document.write( "Use the power rule, d/dx(x^n) = n x^(n-1), where n = -2: d/dx(1/x^2) = d/dx(x^(-2)) = -2 x^(-3):<BR>\n" );
document.write( "f'''(x) = -2 (-2)/x^3<BR>\n" );
document.write( "Simplify the expression:<BR>\n" );
document.write( "f'''(x) = 4/x^3<BR>\n" );
document.write( "Expand the left hand side:<BR>\n" );
document.write( "f'''(x) = 4/x^3<BR>\n" );
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document.write( "evaluate<BR>\n" );
document.write( "f'''(-3) = 4 / (-3)^3 = -4 / 27 = -0.148148148 approx -0.15<BR>\n" );
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