document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=343519>Question 343519</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Obtain dy/dx (rate of change) using the direct rule:<BR>\n" );
document.write( "y= 2.5x^4 - x^3/2.5 - x/2.5 +2.5<BR>\n" );
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document.write( "y= (x^3-2.5x^2)/(2.5sqrtx) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #245835 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=245835\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "I'm unfamiliar with the term \"direct rule.\"  For the first function you need repeated application of the power rule (presuming that I am interpreting your rendering correctly)\r<BR>\n" );
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document.write( "<i><b>Power Rule</b></i>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}x^n\ =\ nx^{n-1}\">\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\left(2.5x^4\ -\ \frac{x^3}{2.5}\ -\ \frac{x}{2.5}\ +\ 2.5\right)\ =\ 10x^3\ -\ \frac{6x^2}{5}\ -\ \frac{2}{5}\">\r<BR>\n" );
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document.write( "I'm going to do the second one for you as well because I'm in a good mood.  But next time -- READ THE INSTRUCTIONS -- One question per post.\r<BR>\n" );
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document.write( "For the second one you need the Quotient Rule and the Power Rule:\r<BR>\n" );
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document.write( "<i><b>Quotient Rule</b></i>\r<BR>\n" );
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document.write( "For:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ \frac{f(x)}{g(x)}\">\r<BR>\n" );
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document.write( "where\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ \ \"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE g(x)\ \neq\ 0\"> are both differentiable,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho'(x)\ =\ \frac{f'(x)g(x)\ -\ f(x)g'(x)}{\left[g(x)\right]^2}\">\r<BR>\n" );
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document.write( "And we want to calculate:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\left[\frac{x^3\ -\ 2.5x^2}{2.5sqrt{x}}\right]\">\r<BR>\n" );
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document.write( "Begin with calculating the derivatives of numerator and denominator separately:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f(x)\ =\ x^3\ -\ 2.5x^2\">\r<BR>\n" );
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document.write( "hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ f'(x)\ =\ 3x^2\ -\ 5x\">\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 \ \ \ \ \ \ \ \ \ \ g(x)\ =\ 2.5\sqrt{x}\ =\ 2.5x^{\frac{1}{2}\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ g'(x)\ =\ 1.25x^{-\frac{1}{2}}\ =\ \frac{1.25}{x^{\frac{1}{2}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\left[\frac{x^3\ -\ 2.5x^2}{2.5sqrt{x}}\right]\ =\ \frac{\left[\left(3x^2\ -\ 5x\right)\left(2.5x^{\frac{1}{2}}\right)\right]\ -\ \left[\left(x^3\ -\ 2.5x^2\right)\left(\frac{1.25}{x^{\frac{1}{2}}}\right)\right]}{\left[2.5x^{\frac{1}{2}}\right]^2}\ =\ \frac{\left(7.5x^{\frac{5}{2}}\ -\ 12.5x^{\frac{3}{2}}\right)\ -\ \left(1.25x^{\frac{5}{2}}\ -\ 3.125x^{\frac{3}{2}}\right)}{6.25x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d}{dx}\left[\frac{x^3\ -\ 2.5x^2}{2.5sqrt{x}}\right]\ =\ \frac{\left(6x^{\frac{3}{2}}\ -\ 10x^{\frac{1}{2}}\right)\ -\ \left(x^{\frac{3}{2}}\ -\ 2.5x^{\frac{1}{2}}\right)}{5}\">\r<BR>\n" );
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document.write( "John<BR>\n" );
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document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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