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      <text>11.2.1.2 Key Point</text>
    </name>
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      <text><![CDATA[<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>Particularly important is the rule for differentiating powers of functions:</p>
<p></p>
If 
\[y = x^n \hspace{2cm} \text{then} \hspace{2cm} \dfrac{dy}{dx} = nx^{n-1} \]
</div>

For example, if \(y = x^3\) then \( \dfrac{dy}{dx} = 3x^2\).

<hr>
<h4 class="HELM_example">Example</h4>
<p> Use the table of derivatives to find \( \dfrac{dy}{dx} \) when \(y\) is given by </p>
<ol class="HELM_parts_inline">
<li>\(7x\)</li>
<li>\(14 \)</li>
<li>\(5x^2\)</li>
<li>\(4x^7\)</li>
</ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
  <li>We note that \(7x\) is of the form \(kx\) where \(k = 7\). Using the table we then have
\( \dfrac{dy}{dx} = 7\).</li>
  <li>Noting that \(14\) is a constant we see that \( \dfrac{dy}{dx} = 0\).</li>
  <li>We see that \(5x^2\) is of the form \(kx^n\), with \(k = 5\) and \(n = 2\). The derivative, \(knx^{n-1}\), is then \(10x^1\), or more simply, \(10x\). So if \(y = 5x^2\), then \( \dfrac{dy}{dx}
= 10x\).</li>
<li>We see that \(4x^7\) is of the form \(kx^n\), with \(k = 4\) and \(n = 7\). Hence the derivative, \( \dfrac{dy}{dx}\), is given by \(28x^6\).</li></ol>]]></text>
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