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310 lines
13 KiB
HTML
310 lines
13 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Introduction to Derivatives</h1>
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<p class="center">It is all about slope!</p>
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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 class="large">
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<p class="center large">Slope = <span class="intbl"><em>Change in Y</em><strong>Change in X</strong></span></p></td>
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<td style="width:30px;"> </td>
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<td><img src="../algebra/images/slope.svg" alt="gradient"></td>
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</tr>
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</tbody></table>
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<p> </p>
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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>
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<p>We can find an <b>average</b> slope between two points.</p>
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<p> </p></td>
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<td> </td>
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<td><img src="images/slope-average.svg" alt="average slope = 24/15"></td>
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</tr>
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<tr>
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<td>
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<p>But how do we find the slope <b>at a point</b>?</p>
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<p>There is nothing to measure!</p></td>
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<td> </td>
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<td><img src="images/slope-0-0.svg" alt="slope 0/0 = ????"></td>
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</tr>
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<tr>
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<td>
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<p>But with derivatives we use a small difference ...</p>
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<p class="center">... then have it <b>shrink towards zero</b>.</p></td>
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<td> </td>
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<td><img src="images/slope-dy-dx.svg" alt="slope delta y / delta x"></td>
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</tr>
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</tbody></table>
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<h2>Let us Find a Derivative!</h2>
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<p>To find the derivative of a function <span class="large">y = f(x)</span> we use the slope formula:</p>
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<p class="center large">Slope = <span class="intbl">
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<em>Change in Y</em>
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<strong>Change in X</strong>
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</span> = <span class="intbl"><em>Δy</em><strong>Δx</strong></span></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>And (from the diagram) we see that:</p>
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<table align="center" cellpadding="3" border="0">
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<tbody>
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<tr>
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<td><span class="center">x changes from</span></td>
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<td> </td>
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<td style="text-align:center;"><span class="center"><span class="large">x</span></span></td>
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<td style="text-align:center;">to</td>
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<td style="text-align:center;"><span class="center"><span class="large">x+Δx</span></span></td>
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</tr>
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<tr>
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<td><span class="center">y changes from</span></td>
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<td> </td>
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<td style="text-align:center;"><span class="center"><span class="large">f(x)</span></span></td>
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<td style="text-align:center;">to</td>
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<td style="text-align:center;"><span class="center"><span class="large">f(x+Δx)</span></span></td>
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</tr>
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</tbody></table>
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<p>Now follow these steps:</p>
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<ul>
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<li>Fill in this slope formula: <span class="intbl large">
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<em>Δy</em><strong>Δx</strong></span> = <span class="intbl large"><em>f(x+Δx) − f(x)</em><strong>Δx</strong></span></li>
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<li>Simplify it as best we can</li>
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<li>Then make <span class="large"><b>Δx</b></span> shrink towards zero.</li>
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</ul>
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<div style="clear:both"></div>
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<p>Like this:</p>
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<div class="example">
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<h3>Example: the function <b>f(x) = x<sup>2</sup></b></h3>
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<p>We know <b>f(x) = x<sup>2</sup></b>, and we can calculate <b>f(x<span class="large">+Δx</span>)</b> :</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;">Start with:</td>
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<td> </td>
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<td><b>f(x<span class="large">+Δx</span>) = (x<span class="large">+Δx</span>)<sup>2</sup></b></td>
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</tr>
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<tr>
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<td style="text-align:right;"><a href="../algebra/expanding.html">Expand</a> (x + Δx)<sup>2</sup>: </td>
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<td> </td>
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<td><b>f(x<span class="large">+Δx</span>) = x<sup>2</sup> + 2x Δx + (Δx)<sup>2</sup></b></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">The slope formula is:</span><span class="right"><span class="intbl">
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<em>f(x+Δx) − f(x)</em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Put in <b>f(x+Δx)</b> and <b>f(x)</b>:</span><span class="right"><span class="intbl">
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<em>x<sup>2</sup> + 2x Δx + (Δx)<sup>2</sup> − x<sup>2</sup></em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Simplify (x<sup>2</sup> and −x<sup>2</sup> cancel):</span><span class="right"><span class="intbl">
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<em>2x Δx + (Δx)<sup>2</sup></em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Simplify more (divide through by <span class="large">Δx)</span>:</span><span class="right">= 2x + Δx</span></div>
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<div class="row"><span class="left">Then, <b>as <span class="large">Δx</span> heads towards 0</b> we get:</span><span class="right">= 2x</span></div>
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</div>
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<p> </p>
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<p class="larger">Result: the derivative of <b> x<sup>2</sup></b> is <b>2x</b></p>
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<p>In other words, the slope at x is <b>2x</b></p>
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</div>
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<p> </p>
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<div class="def">
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<p>We write <b>dx</b> instead of <b>"Δx heads towards 0"</b>.</p>
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<p>And "the derivative of" is commonly written <span class="intbl large"><em>d</em><strong>dx</strong></span> like this:</p>
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<p class="center"><span class="larger"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>2</sup> = 2x</span><br>
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<i>"The derivative of <b>x<sup>2</sup></b> equals <b>2x</b>"</i><br>
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or simply<i> "d dx of <b>x<sup>2</sup></b> equals <b>2x</b>"</i></p>
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</div>
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<br>
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<h3>So what does <b><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>2</sup> = 2x</b> mean?</h3>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/slope-x2-2.svg" alt="slope x^2 at 2 is 4"></p>
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<p>It means that, for the function x<sup>2</sup>, the slope or "rate of change" at any point is <span class="center"> <b>2x</b>.</span></p>
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<p>So when <b>x=2</b> the slope is <b>2x = 4</b>, as shown here:</p>
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<p>Or when <b>x=5</b> the slope is <b>2x = 10</b>, and so on.</p>
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<div class="def">
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<p>Note: <span class="large">f’(x)</span> can also be used for "the derivative of":</p>
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<p class="center"><span class="larger">f’(x) = 2x</span><br>
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<i>"The derivative of f(x) equals 2x"</i><br>
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or simply <i>"f-dash of x equals 2x"</i></p>
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</div>
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<p> </p>
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<p>Let's try another example.</p>
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<div class="example">
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<h3>Example: What is <span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>3</sup> ?</h3>
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<p>We know <b>f(x) = x<sup>3</sup></b>, and can calculate <b>f(x+Δx)</b> :</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;">Start with:</td>
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<td> </td>
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<td><b>f(x+Δx) = (x+Δx)<sup>3</sup></b></td>
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</tr>
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<tr>
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<td style="text-align:right;"><a href="../algebra/expanding.html">Expand</a> (x + Δx)<sup>3</sup>: </td>
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<td> </td>
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<td nowrap="nowrap"><b>f(x+Δx) = x<sup>3</sup> + 3x<sup>2</sup> Δx + 3x (Δx)<sup>2</sup> + (Δx)<sup>3</sup></b></td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">The slope formula:</span><span class="right"><span class="intbl">
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<em>f(x+Δx) − f(x)</em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Put in <b>f(x+Δx)</b> and <b>f(x)</b>:</span><span class="right"><span class="intbl">
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<em>x<sup>3</sup> + 3x<sup>2</sup> Δx + 3x (Δx)<sup>2</sup> + (Δx)<sup>3</sup> − x<sup>3</sup></em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Simplify (x<sup>3</sup> and −x<sup>3</sup> cancel):</span><span class="right"><span class="intbl">
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<em>3x<sup>2</sup> Δx + 3x (Δx)<sup>2</sup> + (Δx)<sup>3</sup></em>
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<strong>Δx</strong>
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</span></span></div>
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<div class="row"><span class="left">Simplify more (divide through by <span class="large">Δx)</span>:</span><span class="right"> 3x<sup>2</sup> + 3x Δx + (Δx)<sup>2</sup></span></div>
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<div class="row"><span class="left">Then, <b>as <span class="large">Δx</span> heads towards 0</b> we get:</span><span class="right">3x<sup>2</sup></span></div>
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</div>
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<p class="larger">Result: the derivative of <b> x<sup>3</sup></b> is <b>3x<sup>2</sup></b></p>
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</div>
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<p>Have a play with it using the <a href="derivative-plotter.html">Derivative Plotter</a>.</p>
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<p> </p>
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<h2>Derivatives of Other Functions</h2>
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<p>We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).</p>
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<div class="center80">
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<p class="center">But <b>in practice</b> the usual way to find derivatives is to use:</p>
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<p class="center large"><a href="derivatives-rules.html">Derivative Rules</a></p>
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</div><p> </p>
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<div class="example">
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<h3>Example: what is the derivative of sin(x) ?</h3>
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<p>On <a href="derivatives-rules.html">Derivative Rules</a> it is listed as being <span class="large">cos(x)</span></p>
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<p>Done.</p>
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</div>
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<p>But using the rules can be tricky!</p>
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<div class="example">
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<h3>Example: what is the derivative of cos(x)sin(x) ?</h3>
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<p>We get a <b>wrong </b>answer if we try to multiply the derivative of cos(x) by the derivative of sin(x) ... !</p>
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<p>Instead we use the "Product Rule" as explained on the <a href="derivatives-rules.html">Derivative Rules</a> page.</p>
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<p>And it actually works out to be <span class="large">cos<sup>2</sup>(x) − sin<sup>2</sup>(x)</span></p>
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</div>
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<p>So that is your next step: learn how to use the rules.</p>
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<p> </p>
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<h2>Notation</h2>
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<p>"Shrink towards zero" is actually written as a <a href="limits.html">limit</a> like this:</p>
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<div class="center larger">f’(x) = <span class="lim"><em>lim</em><strong>Δx→0</strong></span> <span class="intbl"><em>f(x+Δx) − f(x)</em><strong>Δx</strong></span></div>
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<!-- f'(x) = LIM[DELx->0] f(x+DELx)-f(x)/DELx -->
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<p class="center">
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"The derivative of <b>f</b> equals <br><b>the limit as <span class="large">Δ</span>x goes to zero</b> of <span class="large">f(x+Δx) - f(x) over </span><span class="large">Δx</span>"</p>
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<p> </p>
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<p>Or sometimes the derivative is written like this<span class="center"> (explained on <a href="derivatives-dy-dx.html">Derivatives as dy/dx</a></span>):</p>
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<div class="center larger"><span class="intbl"><em>dy</em><strong>dx</strong></span> = <span class="intbl"><em>f(x+dx) − f(x)</em><strong>dx</strong></span></div>
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<!-- dy/dx = f(x+dx)-f(x)/dx -->
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<p> </p>
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<p>The process of finding a derivative is called "differentiation".</p>
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<div class="words">
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<p>You <b>do</b> differentiation ... to <b>get</b> a derivative.</p>
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</div>
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<h2>Where to Next?</h2>
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<p>Go and learn how to find derivatives using <a href="derivatives-rules.html">Derivative Rules</a>, and get plenty of practice:</p>
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<div class="questions">
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<script>getQ(6790, 6791, 6792, 6793, 6794, 6795, 6796, 6797, 6798, 6799);</script>
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||
</div>
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<div class="related">
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