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229 lines
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<title>Derivatives as dy/dx</title>
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Derivatives as dy/dx</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/slope-dy-dx.svg" alt="slope delta y / delta x"></p>
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<p> </p>
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<p>Derivatives are all about <b>change</b> ...</p>
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<div class="indent50px">... they show how fast something is changing (called the <b>rate of change</b>) at any point.</div>
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<p> </p>
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<p>In <a href="derivatives-introduction.html">Introduction to Derivatives</a> <i>(please read it first!)</i> we looked at how to do a derivative using <b>differences</b> and <b>limits</b>.</p>
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<p>Here we look at doing the same thing but using the "dy/dx" notation (also called <i>Leibniz's notation</i>) instead of limits.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/slope-dy-dx2.svg" alt="slope delta x and delta y"></p>
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<p> </p>
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<p>We start by calling the function "y":</p>
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<p class="center large">y = f(x)</p>
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<h2>1. Add Δx</h2>
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<p>When x increases by Δx, then y increases by Δy :</p>
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<p class="center larger">y + Δy = f(x + Δx)</p>
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<h2>2. Subtract the Two Formulas</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>From:</td>
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<td> </td>
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<td class="large" align="center">y + Δy = f(x + Δx) </td>
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</tr>
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<tr>
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<td>Subtract:</td>
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<td> </td>
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<td class="large" align="center">y = f(x) </td>
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</tr>
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<tr>
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<td>To Get:</td>
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<td> </td>
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<td class="large" style="border-top: 1px solid black;" align="center"> y + Δy − y = f(x + Δx) − f(x) </td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td>Simplify:</td>
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<td> </td>
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<td class="larger" align="center">Δy = f(x + Δx) − f(x) </td>
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</tr>
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</tbody></table>
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<h2>3. Rate of Change</h2>
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<p>To work out how fast (called the <b>rate of change</b>) we <b>divide by Δx</b>:</p>
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<p class="center larger"><span class="intbl"><em>Δy</em><strong>Δx</strong></span> = <span class="intbl"><em>f(x + Δx) − f(x)</em><strong>Δx</strong></span></p>
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<h2>4. Reduce Δx close to 0</h2>
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<p>We can't let Δx become 0 (because that would be dividing by 0), but we can make it <b>head towards zero</b> and call it "dx":</p>
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<p class="center large">Δx <img src="../images/style/right-arrow.gif" alt="right arrow" align="absmiddle" height="46" width="46"> dx</p>
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<p>You can also think of "dx" as being <b>infinitesimal</b>, or infinitely small.</p>
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<p>Likewise Δy becomes very small and we call it "dy", to give us:</p>
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<p class="center larger"><span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span> = <span class="intbl">
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<em>f(x + dx) − f(x)</em>
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<strong>dx</strong>
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</span></p>
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<h2>Try It On A Function</h2>
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<p>Let's try f(x) = x<sup>2</sup></p>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td><span class="large"><span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span></span></td>
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<td><span class="large">= <span class="intbl">
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<em>f(x + dx) − f(x)</em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl">
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<em>(x + dx)<sup>2</sup> − x<sup>2</sup></em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td>f(x) = x<sup>2</sup></td>
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</tr>
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<tr>
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<td> </td>
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<td style="white-space:nowrap"><span class="large">= <span class="intbl">
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<em>x<sup>2</sup> + 2x(dx) + (dx)<sup>2</sup> − x<sup>2</sup></em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td>Expand (x+dx)<sup>2</sup></td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl">
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<em>2x(dx) + (dx)<sup>2</sup></em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td>x<sup>2</sup>−x<sup>2</sup>=0</td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= 2x + dx </span></td>
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<td> </td>
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<td>Simplify fraction</td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= 2x</span></td>
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<td> </td>
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<td>dx goes towards 0</td>
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</tr>
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</tbody></table>
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<p>So the derivative of <b>x<sup>2</sup></b> is <b>2x</b></p>
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<p> </p>
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<div class="fun">
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<h3>Why don't you try it on f(x) = x<sup>3</sup> ?</h3>
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<table align="center" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td><span class="large"><span class="intbl">
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<em>dy</em>
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<strong>dx</strong>
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</span></span></td>
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<td><span class="large">= <span class="intbl">
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<em>f(x + dx) − f(x)</em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td> </td>
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<td><span class="large">= <span class="intbl">
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<em>(x + dx)<sup>3</sup> − x<sup>3</sup></em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td>f(x) = x<sup>3</sup></td>
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</tr>
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<tr>
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<td> </td>
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<td style="white-space:nowrap"><span class="large">= <span class="intbl">
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<em>x<sup>3</sup> + ... (your turn!)</em>
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<strong>dx</strong>
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</span></span></td>
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<td> </td>
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<td>Expand (x+dx)<sup>3</sup></td>
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</tr>
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</tbody></table>
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<p>What derivative do <i>you</i> get?</p>
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</div>
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<p> </p>
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<p> </p>
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<div class="related">
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<a href="index.html">Calculus Index</a>
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<a href="limits.html">Introduction to Limits</a>
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