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534 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Derivative Rules</h1>
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<p class="center"><i>The <a href="derivatives-introduction.html">Derivative</a> tells us the slope of a function at any point.</i></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/slope-examples.svg" alt="slope examples y=3, slope=0; y=2x, slope=2" style="width:139px; height:250px; min-width:139px;"></p>
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<p>There are <b>rules</b> we can follow to find many derivatives.</p>
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<p>For example:</p>
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<ul>
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<li>The slope of a <b>constant</b> value (like 3) is always 0</li>
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<li>The slope of a <b>line</b> like 2x is 2, or 3x is 3 etc</li>
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<li>and so on.</li>
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</ul>
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<p>Here are useful rules to help you work out the derivatives of many functions (with <a href="#examples">examples below</a>). Note: the little mark <span class="hilite">’</span> means <b>derivative of</b>, and f and g are functions.</p>
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<div style="clear:both"></div>
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<div class="beach">
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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>Common Functions</th>
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<th align="center" width="120">Function<br>
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</th>
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<th align="center" width="120">Derivative<br>
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</th>
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</tr>
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<tr>
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<td>Constant</td>
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<td style="text-align:center;">c</td>
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<td style="text-align:center;">0</td>
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</tr>
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<tr>
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<td>Line</td>
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<td style="text-align:center;">x</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> </td>
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<td style="text-align:center;">ax</td>
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<td style="text-align:center;">a</td>
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</tr>
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<tr>
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<td>Square</td>
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<td style="text-align:center;">x<sup>2</sup></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>Square Root</td>
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<td style="text-align:center;">√x</td>
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<td style="text-align:center;">(½)x<sup>-½</sup></td>
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</tr>
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<tr>
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<td>Exponential</td>
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<td style="text-align:center;">e<sup>x</sup></td>
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<td style="text-align:center;">e<sup>x</sup></td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;">a<sup>x</sup></td>
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<td style="text-align:center;">ln(a) a<sup>x</sup></td>
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</tr>
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<tr>
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<td>Logarithms</td>
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<td style="text-align:center;">ln(x)</td>
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<td style="text-align:center;">1/x</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;">log<sub>a</sub>(x)</td>
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<td style="text-align:center;">1 / (x ln(a))</td>
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</tr>
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<tr>
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<td>Trigonometry (x is in <a href="../geometry/radians.html">radians</a>)</td>
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<td style="text-align:center;">sin(x)</td>
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<td style="text-align:center;">cos(x)</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;">cos(x)</td>
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<td style="text-align:center;">−sin(x)</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;">tan(x)</td>
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<td style="text-align:center;">sec<sup>2</sup>(x)</td>
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</tr>
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<tr>
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<td>Inverse Trigonometry</td>
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<td style="text-align:center;">sin<sup>-1</sup>(x)</td>
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<td style="text-align:center;">1/√(1−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="text-align:center;">cos<sup>-1</sup>(x)</td>
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<td style="text-align:center;">−1/√(1−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="text-align:center;">tan<sup>-1</sup>(x)</td>
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<td style="text-align:center;">1/(1+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="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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<th>Rules</th>
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<th align="center">Function<br>
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</th>
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<th align="center">Derivative<br>
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</th>
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</tr>
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<tr>
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<td>Multiplication by constant</td>
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<td style="text-align:center;">cf</td>
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<td style="text-align:center;">cf’</td>
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</tr>
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<tr>
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<td><a href="power-rule.html">Power Rule</a></td>
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<td style="text-align:center;">x<sup>n</sup></td>
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<td style="text-align:center;">nx<sup>n−1</sup></td>
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</tr>
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<tr>
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<td>Sum Rule</td>
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<td style="text-align:center;">f + g</td>
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<td style="text-align:center;">f’ + g’</td>
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</tr>
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<tr>
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<td>Difference Rule</td>
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<td style="text-align:center;">f - g</td>
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<td style="text-align:center;">f’ − g’</td>
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</tr>
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<tr>
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<td><a href="product-rule.html">Product Rule</a></td>
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<td style="text-align:center;">fg</td>
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<td style="text-align:center;">f g’ + f’ g</td>
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</tr>
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<tr>
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<td>Quotient Rule</td>
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<td style="text-align:center;">f/g</td>
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<td style="text-align:center;"><span class="intbl"><em>f’ g − g’ f</em><strong>g<sup>2</sup></strong></span></td>
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</tr>
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<tr>
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<td>Reciprocal Rule</td>
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<td style="text-align:center;">1/f</td>
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<td style="text-align:center;">−f’/f<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="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>Chain Rule<br>
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(as <a href="../sets/functions-composition.html">"Composition of Functions")</a></td>
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<td style="text-align:center;">f º g</td>
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<td style="text-align:center;">(f’ º g) × g’</td>
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</tr>
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<tr>
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<td>Chain Rule
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(using ’ )</td>
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<td style="text-align:center;">f(g(x))</td>
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<td style="text-align:center;">f’(g(x))g’(x)</td>
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</tr>
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<tr>
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<td>Chain Rule
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(using <span class="intbl">
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<em>d</em>
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<strong>dx</strong>
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</span> )</td>
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<td colspan="2" align="center"><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>dy</em>
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<strong>du</strong>
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</span><span class="intbl">
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<em>du</em>
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<strong>dx</strong>
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</span></td>
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</tr>
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</tbody></table>
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<div class="words">
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<p>"The derivative of" is also written <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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<p>So <span 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>sin(x)</span> and <span class="center large">sin(x)’</span> both mean "The derivative of sin(x)"</p>
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</div>
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</div>
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<h2><a name="examples"></a>Examples</h2>
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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>From the table above it is listed as being <b>cos(x)</b></p>
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<p>It can be written as:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = cos(x)</p>
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<p>Or:</p>
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<p class="center large">sin(x)’ = cos(x)</p>
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</div>
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<h3>Power Rule</h3>
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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>The question is asking "what is the derivative of x<sup>3</sup> ?"</p>
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<p>We can use the <a href="power-rule.html">Power Rule</a>, where n=3:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>n</sup> = nx<sup>n−1</sup></p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>3</sup> = 3x<sup>3−1</sup> = <b>3x<sup>2</sup></b></p>
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<p>(In other words the derivative of x<sup>3</sup> is 3x<sup>2</sup>)</p>
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</div>
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<p>So it is simply this:</p>
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<p class="center"><img src="images/power-rule.svg" alt="power rule x^3 -> 3x^2" style="width:66px; height:107px; min-width:66px;"><br>
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"multiply by power<br>
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then reduce power by 1"</p>
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<p>It can also be used in cases like this:</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>(1/x) ?</h3>
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<p>1/x is also <b>x<sup>-1</sup></b></p>
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<p>We can use the Power Rule, where n = −1:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>n</sup> = nx<sup>n−1</sup></p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>-1</sup> = −1x<sup>-1−1</sup></p>
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<p class="center large">= −x<sup>-2</sup></p>
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<p class="center large">= <span class="intbl"><em>−1</em><strong>x<sup>2</sup></strong></span></p>
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</div>
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<p>So we just did this:</p>
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<p class="center"><img src="images/power-rule-1.svg" alt="power rule x^-1 -> -x^-2" style="width:73px; height:107px; min-width:73px;"><br>
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which simplifies to <b>−1/x<sup>2</sup></b></p>
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<h3>Multiplication by constant</h3>
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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>5x<sup>3 </sup>?</h3>
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<p class="center large">the derivative of cf = cf’</p>
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<p class="center large">the derivative of 5f = 5f’</p>
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<p>We know (from the Power Rule):</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>3</sup> = 3x<sup>3−1</sup> = 3x<sup>2</sup></p>
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<p>So:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>5x<sup>3</sup> = 5<span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>3</sup> = 5 × 3x<sup>2</sup> = <b>15x<sup>2</sup></b></p>
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</div>
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<h3>Sum Rule</h3>
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<div class="example">
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<h3>Example: What is the derivative of x<sup>2</sup>+x<sup>3 </sup>?</h3>
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<p>The Sum Rule says:</p>
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<p class="center large">the derivative of f + g = f’ + g’</p>
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<p>So we can work out each derivative separately and then add them.</p>
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<p>Using the Power Rule:</p>
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<ul>
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<li><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>2</sup> = 2x</li>
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<li><span class="intbl"><em>d</em><strong>dx</strong></span>x<sup>3</sup> = 3x<sup>2</sup></li>
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</ul>
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<p>And so:</p>
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<p class="center large">the derivative of x<sup>2</sup> + x<sup>3</sup> = <b>2x + 3x<sup>2</sup></b></p>
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</div>
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<h3>Difference Rule</h3>
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<p>What we differentiate with respect to doesn't have to be <b>x</b>, it could be anything. In this case <b>v</b>:</p>
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<div class="example">
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<h3>Example: What is <span class="intbl"><em>d</em><strong>dv</strong></span>(v<sup>3</sup>−v<sup>4</sup>) ?</h3>
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<p>The Difference Rule says</p>
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<p class="center large">the derivative of f − g = f’ − g’</p>
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<p>So we can work out each derivative separately and then subtract them.</p>
|
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|
||
<p>Using the Power Rule:</p>
|
||
<ul>
|
||
<li><span class="intbl"><em>d</em><strong>dv</strong></span>v<sup>3</sup> = 3v<sup>2</sup></li>
|
||
|
||
<li><span class="intbl"><em>d</em><strong>dv</strong></span>v<sup>4</sup> = 4v<sup>3</sup></li>
|
||
</ul>
|
||
<p>And so:</p>
|
||
<p class="center large">the derivative of v<sup>3</sup> − v<sup>4</sup> = <b> 3v<sup>2</sup> − 4v<sup>3</sup></b></p>
|
||
</div>
|
||
|
||
<h3>Sum, Difference, Constant Multiplication And Power Rules</h3>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl"><em>d</em><strong>dz</strong></span>(5z<sup>2</sup> + z<sup>3</sup> − 7z<sup>4</sup>) ?</h3>
|
||
<p>Using the Power Rule:</p>
|
||
<ul>
|
||
<li><span class="intbl"><em>d</em><strong>dz</strong></span>z<sup>2</sup> = 2z</li>
|
||
<li><span class="intbl"><em>d</em><strong>dz</strong></span>z<sup>3</sup> = 3z<sup>2</sup></li>
|
||
<li><span class="intbl"><em>d</em><strong>dz</strong></span>z<sup>4</sup> = 4z<sup>3</sup></li>
|
||
</ul>
|
||
<p>And so:</p>
|
||
<p class="center large"><span class="intbl"><em>d</em><strong>dz</strong></span>(5z<sup>2</sup> + z<sup>3</sup> − 7z<sup>4</sup>) = 5 × 2z + 3z<sup>2</sup> − 7 × 4z<sup>3</sup><br>
|
||
= <b>10z + 3z<sup>2</sup> − 28z<sup>3</sup></b></p>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Product Rule</h3>
|
||
<div class="example">
|
||
<h3>Example: What is the derivative of cos(x)sin(x) ?</h3>
|
||
<p>The Product Rule says:</p>
|
||
<p class="center large">the derivative of fg = f g’ + f’ g</p>
|
||
<p>In our case:</p>
|
||
<ul>
|
||
<li>f = cos</li>
|
||
<li>g = sin</li>
|
||
</ul>
|
||
<p>We know (from the table above):</p>
|
||
<ul>
|
||
<li><span class="intbl"><em>d</em><strong>dx</strong></span>cos(x) = −sin(x)</li>
|
||
<li><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = cos(x)</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large">the derivative of cos(x)sin(x) = cos(x)cos(x) − sin(x)sin(x)<br>
|
||
<br>
|
||
= <b>cos<sup>2</sup>(x) − sin<sup>2</sup>(x)</b></p>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Quotient Rule</h3>
|
||
<p>To help you remember:</p>
|
||
|
||
<p class="center large">(<span class="intbl"><em>f</em><strong>g</strong></span>)’ = <span class="intbl"><em>gf’ − fg’</em><strong>g<sup>2</sup></strong></span></p>
|
||
|
||
<p>The derivative of "High over Low" is:</p>
|
||
<p class="center"><i><span class="large">"Low dHigh minus High dLow, over the line and square the Low"</span></i></p>
|
||
<div class="example">
|
||
<h3>Example: What is the derivative of cos(x)/x ?</h3>
|
||
<p>In our case:</p>
|
||
<ul>
|
||
<li>f = cos</li>
|
||
<li>g = x</li>
|
||
</ul>
|
||
<p>We know (from the table above):</p>
|
||
<ul>
|
||
<li>f' = −sin(x)</li>
|
||
<li>g' = 1</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large">the derivative of <span class="intbl"><em>cos(x)</em><strong>x</strong></span> = <span class="intbl"><em>Low dHigh minus High dLow</em><strong>square the Low</strong></span></p>
|
||
<p class="center large">= <span class="intbl"><em>x(−sin(x)) − cos(x)(1)</em><strong>x<sup>2</sup></strong></span></p>
|
||
<p class="center large">= −<span class="intbl"><em>xsin(x) + cos(x)</em><strong>x<sup>2</sup></strong></span></p>
|
||
</div>
|
||
<p> </p>
|
||
<h3>Reciprocal Rule</h3>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl"><em>d</em><strong>dx</strong></span>(1/x) ?</h3>
|
||
<p>The Reciprocal Rule says:</p>
|
||
<p class="center large">the derivative of <span class="intbl"><em>1</em><strong>f</strong></span> = <span class="intbl"><em>−f’</em><strong>f<sup>2</sup></strong></span></p>
|
||
<p><b>With f(x)= x, we know that f’(x) = 1</b></p>
|
||
<p>So:</p>
|
||
<p class="center large">the derivative of <span class="intbl"><em>1</em><strong>x</strong></span> = <span class="intbl"><em>−1</em><strong>x<sup>2</sup></strong></span></p>
|
||
<p>Which is the same result we got above using the Power Rule.</p>
|
||
</div>
|
||
<h3>Chain Rule</h3>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span>sin(x<sup>2</sup>) ?</h3>
|
||
<p><b>sin(x<sup>2</sup>)</b> is made up of <b>sin()</b> and <b>x<sup>2</sup></b>:</p>
|
||
<ul>
|
||
<li>f(g) = sin(g)</li>
|
||
<li>g(x) = x<sup>2</sup></li>
|
||
</ul>
|
||
<p>The Chain Rule says:</p>
|
||
<p class="center large">the derivative of f(g(x)) = f'(g(x))g'(x)</p>
|
||
<p>The individual derivatives are:</p>
|
||
<ul>
|
||
<li>f'(g) = cos(g)</li>
|
||
<li>g'(x) = 2x</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large"><span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span>sin(x<sup>2</sup>) = cos(g(x)) (2x)</p>
|
||
<p class="center large">= 2x cos(x<sup>2</sup>)</p>
|
||
</div> <p>Another way of writing the Chain Rule is: <span class="intbl">
|
||
<em>dy</em>
|
||
<strong>dx</strong>
|
||
</span> = <span class="intbl">
|
||
<em>dy</em>
|
||
<strong>du</strong>
|
||
</span><span class="intbl">
|
||
<em>du</em>
|
||
<strong>dx</strong>
|
||
</span></p>
|
||
<p>Let's do the previous example again using that formula:</p>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span>sin(x<sup>2</sup>) ?</h3>
|
||
<p class="center"><span class="intbl">
|
||
<em>dy</em>
|
||
<strong>dx</strong>
|
||
</span> = <span class="intbl">
|
||
<em>dy</em>
|
||
<strong>du</strong>
|
||
</span><span class="intbl">
|
||
<em>du</em>
|
||
<strong>dx</strong>
|
||
</span></p>
|
||
<p>Let u = x<sup>2</sup>, so y = sin(u):</p>
|
||
<p class="center"><span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong></span> sin(x<sup>2</sup>) = <span class="intbl">
|
||
<em>d</em>
|
||
<strong>du</strong>
|
||
</span>sin(u)<span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span>x<sup>2</sup></p>
|
||
<p>Differentiate each:</p>
|
||
<p class="center"><span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span> sin(x<sup>2</sup>) = cos(u) (2x)</p>
|
||
<p>Substitute back u = x<sup>2</sup> and simplify:</p>
|
||
<p class="center large"><span class="intbl">
|
||
<em>d</em>
|
||
<strong>dx</strong>
|
||
</span> sin(x<sup>2</sup>) = 2x cos(x<sup>2</sup>)</p>
|
||
<p>Same result as before (thank goodness!)</p>
|
||
</div>
|
||
<p>Another couple of examples of the Chain Rule:</p>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl"><em>d</em><strong>dx</strong></span>(1/cos(x)) ?</h3>
|
||
<p><b>1/cos(x)</b> is made up of <b>1/g</b> and <b>cos()</b>:</p>
|
||
<ul>
|
||
<li>f(g) = 1/g</li>
|
||
<li>g(x) = cos(x)</li>
|
||
</ul>
|
||
<p>The Chain Rule says:</p>
|
||
<p class="center large">the derivative of f(g(x)) = f’(g(x))g’(x)</p>
|
||
<p>The individual derivatives are:</p>
|
||
<ul>
|
||
<li>f'(g) = −1/(g<sup>2</sup>)</li>
|
||
<li>g'(x) = −sin(x)</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large">(1/cos(x))’ = <span class="intbl"><em>−1</em><strong>g(x)<sup>2</sup></strong></span>(−sin(x))</p>
|
||
<p class="center large"><b>= <span class="intbl"><em>sin(x)</em><strong>cos<sup>2</sup>(x)</strong></span></b></p>
|
||
<p>Note: <span class="intbl"><em>sin(x)</em><strong>cos<sup>2</sup>(x)</strong></span> is also <span class="intbl"><em>tan(x)</em><strong>cos(x)</strong></span> or many other forms.</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="example">
|
||
<h3>Example: What is <span class="intbl"><em>d</em><strong>dx</strong></span>(5x−2)<sup>3</sup> ?</h3>
|
||
<p>The Chain Rule says:</p>
|
||
<p class="center large">the derivative of f(g(x)) = f’(g(x))g’(x)</p>
|
||
<p><b>(5x−2)<sup>3</sup></b> is made up of <b>g<sup>3</sup></b> and <b>5x−2</b>:</p>
|
||
<ul>
|
||
<li>f(g) = g<sup>3</sup></li>
|
||
<li>g(x) = 5x−2</li>
|
||
</ul>
|
||
|
||
<p>The individual derivatives are:</p>
|
||
<ul>
|
||
<li>f'(g) = 3g<sup>2</sup> (by the Power Rule)</li>
|
||
<li>g'(x) = 5</li>
|
||
</ul>
|
||
<p>So:</p>
|
||
<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(5x−2)<sup>3</sup> = (3g(x)<sup>2</sup>)(5) = 15(5x−2)<sup>2</sup></p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">6800, 6801, 6802, 6803, 6804, 6805, 6806, 6807, 6808, 6809, 6810, 6811, 6812</div>
|
||
|
||
<div class="related">
|
||
<a href="derivatives-introduction.html">Introduction to Derivatives</a>
|
||
<a href="derivatives-partial.html">Partial Derivatives</a>
|
||
<a href="index.html">Calculus Index</a>
|
||
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|
||
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