document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190590>Question 1190590</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use limits to find the slope of the tangent to f(x) = x^2, at the point (3,9). \r<BR>\n" );
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document.write( "Please use the differentiation from the first principles method if possible, as I don't understand how it would work here. </SPAN>\n" );
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document.write( "f(x) = x^2<BR>\n" );
document.write( "f(x+h) = (x+h)^2 ... every x is replaced with x+h<BR>\n" );
document.write( "f(x+h) = x^2+2xh+h^2 ... expand with the FOIL rule\r<BR>\n" );
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document.write( "Apply the difference quotient<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28%28x%5E2%2B2xh%2Bh%5E2%29-%28x%5E2%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28%28x%5E2%2B2xh%2Bh%5E2%29-%28x%5E2%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28x%5E2%2B2xh%2Bh%5E2-x%5E2%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28x%5E2%2B2xh%2Bh%5E2-x%5E2%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%282xh%2Bh%5E2%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%282xh%2Bh%5E2%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > The x^2 terms cancel out, leaving nothing but terms with h in the numerator\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%282x%2Bh%29%29%2Fh ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28h%282x%2Bh%29%29%2Fh\"   DISABLED_style= \"max-width:90%; height:auto;\" > That way we can factor out h\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28highlight%28h%29%282x%2Bh%29%29%2F%28highlight%28h%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28cross%28h%29%282x%2Bh%29%29%2F%28cross%28h%29%29 ALIGN=MIDDLE  ALT=\"%28f%28x%2Bh%29-f%28x%29%29%2Fh+=+%28cross%28h%29%282x%2Bh%29%29%2F%28cross%28h%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The h's cancel\r<BR>\n" );
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document.write( "Now let h approach 0. It won't actually reach 0 itself, but only get closer and closer. This is the limit process in action. Effectively we replace h with 0 (refer to the substitution limit theorem) and we go from 2x+h to 2x+0 or simply 2x.\r<BR>\n" );
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document.write( "In short, the derivative of f(x) = x^2 is f ' (x) = 2x.<BR>\n" );
document.write( "You'll learn a shortcut (if you haven't already) that's useful and that's the power rule. That will mean you can quickly differentiate any polynomial like this without having to resort to these limits. However it's good to know how all this works out.\r<BR>\n" );
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document.write( "Plug x = 3 into the derivative function to find the tangent slope<BR>\n" );
document.write( "f ' (x) = 2x<BR>\n" );
document.write( "f ' (3) = 2*3<BR>\n" );
document.write( "f ' (3) = 6<BR>\n" );
document.write( "The tangent slope is m = 6\r<BR>\n" );
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document.write( "Answer: <font color=red>6</font><BR>\n" );
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