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<!DOCTYPE html>
<html lang="en-us">
<head>
<title>☆【 Differentiation Website 】☆</title>
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<h3><strong>></strong> DIFFERENTIATION</h3>
<h6>Created by Joanne Lee<span id="blink">|</span></h6>
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<h4><strong>DIRECTORY</strong></h4>
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<ul>
<li><button><a href="index.html" id = "a"><strong>></strong> What is differentiation?</a></button></li>
<li><button><a href="rules.html" id = "b"><strong>></strong> Differentiation rules</a></button></li>
<li><button><a href="game.html" id = "c"><strong>></strong> Game (practice)</a></button></li>
</ul>
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<h3><strong>What is differentiation?</strong></h3>
<span id="colorsquare1">.</span><span id="colorsquare2">.</span><span id="colorsquare3">.</span>
<h5><u>Differentiation</u> refers to the process of finding a derivative. A function that has a derivative at a point is said to be differentiable at that point.</h5>
<p><div id = box4><b>So what is a derivative?</b> A <u>derivative</u> is the slope of a particular curve at a specified point.<br><br>This is also known as:<br><br>
<ul>
<li><b>></b> The slope of the tangent line</li>
<li><b>></b> Instantaneous rate of change</li>
</ul>
If the point of tangency is (a, f(a)), the first derivative is denoted by f'(a).
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<h5 id="white"><i id = "glitch">Definition of the derivative:</i></h5>
<p id = "red">f'(x) = lim <u>f(x + h) - f(x)</u><br> h→0 h<br><br></p>
<p id = "yellow"><b>></b> If f'(x) exists → f is differentiable at x!</p><br>
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<h5 id="white"><i id = "glitch">Alternate form of the derivative:</i></h5>
<p id = "blue">f'(a) = lim <u>f(x) - f(a)</u><br> x→a x-a<br><br></p>
<p id = "yellow"><b>></b> For all <i>a</i> for which the limit exists, f'(x) is a function of a.</p><br>
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<p><b>Okay, but how do we know if a derivative exists at a point?</b>
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f'(a) <u>exists at the point</u> if the left and right limits of the slope are both equal to the same finite value at that point.
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f'(a) <u>fails to exist at the point</u> if there is a corner, a cusp, a vertical tangent line, or a discontinuity.
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<br>
* Differentiability implies continuity!
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