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      <text>10.1.15 (Finding derivatives)</text>
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      <text><![CDATA[<h1>Finding derivatives</h1>

<p>To end this section we will practice how to find various derivatives. We will also examine functions which we cannot find the derivatives of. </p>

<p>We will summarise the main derivative results we have seen so far. For a function \(y = f(x)\) the derivative (with respect to \(x\)) is denoted \(\frac{dy}{dx}\) or \(f'(x)\).</p><p>For the following basic functions we have easy rules for calculating derivatives:<br></p><p><br>
</p><table style="text-align: center; margin-left:auto; margin-right:auto;" cellpadding="10" bordercolor="#3a87ad" border="1" width="100%">
<thead>
<tr style="background: #c1ddea">
<th style="min-width: 300px" scope="col">Function</th>
<th style="min-width: 300px" scope="col">Derivative</th>
</tr>
</thead>
<tbody>
<tr>
<td>\(y = x^n\)</td>
<td>\(\frac{dy}{dx} = n x^{n-1}\)</td>
</tr>
<tr>
<td>\(y= a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0\)</td>
<td>\(\frac{dy}{dx} = na_nx^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + 2a_2x + a_1\)</td>
</tr>
<tr>
<td>\(y = \sin(x)\)</td>
<td>\(\frac{dy}{dx}&nbsp; = \cos(x)\)</td>
</tr>
<tr>
<td>&nbsp;\(y = \cos(x)\)&nbsp;</td>
<td>\(\frac{dy}{dx}&nbsp; = -\sin(x)\)</td>
</tr>
</tbody>
</table><br/>

<h3>Example</h3>

<p>Find the derivative of the function \(y = x^2 + 3x + 2\).</p>

<h3>Solution</h3>

<p>Using the rules above we have that \(\frac{dy}{dx} = 2x + 3\).</p>]]></text>
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