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152 lines
6.4 KiB
HTML
152 lines
6.4 KiB
HTML
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<h1 class="center">Product Rule</h1>
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<p>The product rule tells us the <a href="derivatives-introduction.html">derivative</a> of two functions <b>f</b> and <b>g</b> that are multiplied together:</p>
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<p class="center large">(fg)’ = fg’ + gf’</p>
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<p>(The little mark <span class="hilite">’</span> means "derivative of".)</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 have two functions <b>cos(x)</b> and <b>sin(x)</b> multiplied together, so let's use the Product Rule:</p>
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<p class="center large">(fg)’ = f g’ + f’ g</p>
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<p>Which in our case becomes:</p>
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<p class="center large">(cos(x)sin(x))’ = cos(x) sin(x)’ + cos(x)’ sin(x)</p>
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We know (from <a href="derivatives-rules.html">Derivative Rules</a>) that:
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<ul>
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<li>sin(x)’ = cos(x)</li>
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<li>cos(x)’ = −sin(x)</li>
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</ul>
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<p>So we can substitute:</p>
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<p class="center large">(cos(x)sin(x))’ = cos(x) cos(x) + −sin(x) sin(x)</p>
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<p>Which simplifies to:</p>
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<p class="center large">(cos(x)sin(x))’ = cos<sup>2</sup>(x) − sin<sup>2</sup>(x)</p>
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<p class="center larger"></p>
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Answer: the derivative of cos(x)sin(x) = <b>cos<sup>2</sup>(x) − sin<sup>2</sup>(x)</b>
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</div>
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<h2>Why Does It Work?</h2>
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<p>When we multiply two functions f(x) and g(x) the result is the <b>area fg</b>:</p>
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<p class="center"><img src="images/product-rule.svg" alt="product rule" height="285" width="307"></p>
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<p>The derivative is the rate of change, and when <b>x</b> changes a little then both <b>f</b> and <b>g</b> will also change a little (by Δf and Δg). In this example they both increase making the area bigger.</p>
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<p>How much bigger?</p>
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<p class="center large">Increase in area = Δ(fg) = fΔg + ΔfΔg + gΔf</p>
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<p>As the change in x heads towards zero, the "ΔfΔg" term also heads to zero, and we get:</p>
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<p class="center large">(fg)’ = fg’ + gf’</p>
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<h2>Alternative Notation</h2>
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<p>An alternative way of writing it (called Leibniz Notation) is:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(uv) = u<span class="intbl"><em>dv</em><strong>dx</strong></span> + v<span class="intbl"><em>du</em><strong>dx</strong></span> <span class="intbl"><strong></strong></span><span class="intbl"><strong></strong></span></p>
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<!-- d/dx (uv) = du/dx v + u dv/dx -->
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<p>Here is our example from before in Leibniz Notation:</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>This:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(uv) = u<span class="intbl"><em>dv</em><strong>dx</strong></span> + v<span class="intbl"><em>du</em><strong>dx</strong></span> <span class="intbl"><strong></strong></span></p>
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<p>Becomes this:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(cos(x)sin(x)) = cos(x)<span class="intbl"><em>d(sin(x))</em><strong>dx</strong></span> + sin(x)<span class="intbl"><em>d(cos(x))</em><strong>dx</strong></span><span class="intbl"></span></p>
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From <a href="derivatives-rules.html">Derivative Rules</a>:
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<ul>
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<li><span class="intbl"><em>d</em><strong>dx</strong></span>sin(x) = cos(x)</li>
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<li><span class="intbl"><em>d</em><strong>dx</strong></span>cos(x) = −sin(x)</li>
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</ul> So:
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(cos(x)sin(x)) = cos(x) cos(x) + −sin(x) sin(x)</p>
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<p>Which simplifies to:</p>
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<p class="center large"><span class="intbl"><em>d</em><strong>dx</strong></span>(cos(x)sin(x)) = cos<sup>2</sup>(x) − sin<sup>2</sup>(x)</p>
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</div>
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<h2>Three Functions</h2>
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<p>For three functions multiplied together we can use:</p>
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<p class="center large">(fgh)’ = f’gh + fg’h + fgh’</p>
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<p> </p>
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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="derivatives-introduction.html">Introduction to Derivatives</a>
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<a href="index.html">Calculus Index</a>
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