document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=185259>Question 185259</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Differentiate (find the derivative) of the function 2/(&#8730;x) + x/(&#8730;3) + 6(3&#8730;x <-- cube root over x), the answer should be 1/(&#8730;3) - 1/&#8730;(x^3) - 2/(3&#8730;x^2) <-- cube root over x^2). Please show your steps. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #139001 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=139001\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  f(x) = y = \frac{2}{sqrt{x}} + \frac{x}{sqrt{3}} + \frac{6} {\sqrt[3]{x^2}}\">\r<BR>\n" );
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document.write( "Since we know that:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  \phi(x) = \alpha(x) + \beta(x) \ \ \Rightarrow\ \ \phi'(x) = \alpha'(x) + \beta'(x)\">\r<BR>\n" );
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document.write( "We can take the derivitive of each term of your function and just add them.  First thing to do is put your functions in terms of fractional exponents using the rules:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \sqrt[m]{a^n} = a^{\frac{n}{m}}\">\r<BR>\n" );
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document.write( "and \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1}{a^n} = a^{-n}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ g(x) = \frac{2}{sqrt{x}} \ \ \Rightarrow\ \ g(x) = 2x^{-\frac{1}{2}}\">\r<BR>\n" );
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document.write( "Now take the derivitive using the Power Rule: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{d(ax^n)}{dx} = anx^{n-1}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ g(x) = 2x^{-\frac{1}{2}}\ \ \Rightarrow\ \ g'(x) = 2\left(-\frac{1}{2}\right)x^{-\frac{1}{2}-1}\  = -x^{-\frac{3}{2}}\ =\frac{-1}{sqrt{x^3}} \">\r<BR>\n" );
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document.write( "Look at the next term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ h(x) = \frac{x}{sqrt{3}} \ \ \Rightarrow\ \ h(x) = \frac{1}{sqrt{3}} x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \  h'(x) = \frac{1}{sqrt{3}}\ \cdot\ 1\ x^{1 - 1}\ = \frac{1}{sqrt{3}}x^0\ =  \frac{1}{sqrt{3}} \">\r<BR>\n" );
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document.write( "And then the last term:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k(x) = \frac{6} {\sqrt[3]{x^2}} = 6x^{-\frac{2}{3}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k'(x) = 6\left(-\frac{2}{3}\right)x^{\left(-\frac{2}{3}-1\right)}\  = -4x^{-\frac{5}{3}} = -\frac{4}{\sqrt[3]{x^5}} = -\frac{4}{x\sqrt[3]{x^2}} \">\r<BR>\n" );
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document.write( "Then put it all back together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(x) = g'(x) + h'(x) + k'(x) = \frac{-1}{sqrt{x^3}} + \frac{1}{sqrt{3}} -\frac{4}{x\sqrt[3]{x^2}} \">\r<BR>\n" );
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document.write( "However, I would rationalize all the denominators:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(x) = -\frac{sqrt{x^3}}{x^3} + \frac{sqrt{3}}{3} - \frac{4 \sqrt[3]{x}}{x^2}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
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