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208 lines
7.4 KiB
HTML
208 lines
7.4 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/calculus/power-rule.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:02 GMT -->
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<!-- #BeginEditable "doctitle" -->
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<title>Power Rule</title>
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<body id="bodybg" class="adv">
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<div id="stt"></div>
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<div id="adTop"></div>
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<div id="hdr"></div>
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<div id="cookOK"></div>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Power Rule</h1>
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<p>The Power Rule, one of the most commonly used <a href="derivatives-introduction.html">derivative</a> rules says:</p>
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<div class="def">
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<p class="center">The derivative of <b>x<sup>n</sup></b> is <b>nx<sup>(n−1)</sup></b></p>
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</div>
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<div class="example">
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<h3>Example: What is the derivative of x<sup>2</sup> ?</h3>
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<p>For x<sup>2</sup> we use the Power Rule with <span class="hilite">n=2</span>:</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><span class="center">The derivative of <span class="large">x<sup class="hilite">2</sup></span> </span></td>
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<td> <span class="large">=</span> </td>
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<td><span class="large"><span class="hilite">2</span>x<sup>(<span class="hilite">2</span>−1)</sup></span></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> </td>
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<td><span class="large">2x<sup>1</sup><br>
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</span></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> </td>
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<td><span class="large">2x</span></td>
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</tr>
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</tbody></table>
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<p class="center large">Answer: the derivative of <b>x<sup>2</sup></b> is <b>2x</b></p>
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</div><p> </p>
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<div class="words">
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<p>"The derivative of" can be shown with this little "dash" mark: <b class="hilite">’</b></p>
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</div>
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<p>Using that mark we can write the Power Rule like this:</p>
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<p class="center large">f’(x<sup>n</sup>) = nx<sup>(n−1)</sup></p>
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<div class="example">
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<h3>Example: What is the derivative of x<sup>3</sup> ?</h3>
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<p class="center large">f’(x<sup>3</sup>) = 3x<sup>3−1</sup> = <b>3x<sup>2</sup></b></p>
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</div>
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<div class="words">
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<p>"The derivative of" can also be shown by <span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span></p>
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</div>
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<div class="example">
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<h3>Example: What is <span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span>(1/x) ?</h3>
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<p>1/x is also <b>x<sup>-1</sup></b></p>
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<p>Using the Power Rule with <b>n = −1</b>:</p>
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<p class="center large"><span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span>x<sup>n</sup> = nx<sup>n−1</sup></p>
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<p class="center large"><span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span>x<sup>-1</sup> = −1x<sup>-1−1</sup> = <b>−x<sup>-2</sup></b></p>
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</div>
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<h2>How to Remember</h2>
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<p class="center"><img src="images/power-rule.svg" alt="power rule x^3 -> 3x^2"><br>
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"multiply by power<br>
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then reduce power by 1"</p>
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<div style="clear:both"></div>
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<h2>A Short Table</h2>
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<p>Here is the Power Rule with some sample values. See the pattern?</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<th align="center">f</th>
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<th align="center">f’(x<sup>n</sup>) = nx<sup>(n−1)</sup></th>
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<th align="center">f’</th>
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</tr>
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<tr>
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<td style="text-align:center;">x</td>
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<td style="text-align:center;"><span class="large">1x<sup>(1−1)</sup> = x<sup>0</sup></span></td>
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<td style="text-align:center;">1</td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>2</sup></td>
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<td style="text-align:center;"><span class="large">2x<sup>(2−1)</sup> = 2x<sup>1</sup></span></td>
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<td style="text-align:center;">2x</td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>3</sup></td>
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<td style="text-align:center;"><span class="large">3x<sup>(3−1)</sup> = 3x<sup>2</sup></span></td>
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<td style="text-align:center;">3x<sup>2</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>4</sup></td>
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<td style="text-align:center;"><span class="large">4x<sup>(4−1)</sup> = 4x<sup>3</sup></span></td>
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<td style="text-align:center;">4x<sup>3</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">etc...</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td colspan="3" align="center">And for negative exponents:</td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>-1</sup></td>
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<td style="text-align:center;"><span class="large">−1x<sup>(−1−1)</sup> = −x<sup>-2</sup></span></td>
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<td style="text-align:center;">−x<sup>-2</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>-2</sup></td>
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<td style="text-align:center;"><span class="large">−2x<sup>(−2−1)</sup> = −2x<sup>-3</sup></span></td>
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<td style="text-align:center;">−2x<sup>-3</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">x<sup>-3</sup></td>
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<td style="text-align:center;"><span class="large">−3x<sup>(−3−1)</sup> = −3x<sup>-4</sup></span></td>
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<td style="text-align:center;">−3x<sup>-4</sup></td>
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</tr>
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<tr>
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<td style="text-align:center;">etc...</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;"> </td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(6800, 6801, 6802, 6803, 6804, 6805, 13384, 13385, 13386, 13387);</script>
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</div>
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<div class="related">
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<a href="derivatives-rules.html">Derivative Rules</a>
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<a href="index.html">Calculus Index</a>
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</div>
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