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      <text>10.0.16 (Gradient of a tangent)</text>
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      <text><![CDATA[<h1>Gradient of a tangent</h1>
<p>In this section we will look at how to find the gradient of tangents to curves. We have seen now that this corresponds to an <em>instantaneous rate of change</em>. Drawing a graph and the drawing the appropriate tangent lines can be time consuming and inaccurate. We therefore need a more efficient and accurate method for determining the gradient of a tangent. It transpires that the best way to do this is to follow the graphical method we saw at the start of this week but encode it algebraically. Consider the following graph.</p>
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<p><br /><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Derivative%20gradient%20of%20tangent.png" alt="diagram showing gradient of a tangent" width="558" height="406" /></p>
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<p>We cannot find the gradient of the tangent to a curve \(y=f(x)\) at a specific point A, by direct calculation, because we only know one point of the tangent. However, if B is a second point on the curve, then we have seen that we can find the gradient for the straight line between A and B. This gradient is given by \[ \frac{f(a+h) - f(a)}{h}. \] This quantity is sometimes denoted \(\frac{\delta y}{\delta x}\) where \(\delta y = f(a+h) - f(a)\) and \(\delta x = h\)/ Looking at the graph, we can see that \(\delta y\) represents the vertical step and \(\delta x\) is the horizontal step. These quantities are read as 'delta y' and 'delta x' and they represent the change in \(y\) and the change in \(x\) respectively. To calculate the gradient of the tangent at the point \(A\), we let the point \(B\) get closer and closer to \(A\). This means that the horizontal step, which is \(\delta x = h\) becomes infinitely small, this is where we require the use of limits to formalise this concept.</p>
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<p>For two points \(A(a,f(a))\) and \(B(a+h, f(a+h))\) on a function, we have seen above that the gradient of the chord AB is given by \(\frac{f(a+h) - f(a)}{h}\). Letting the point B become infinitely close (but not equal to) A is equivalent to taking the limit as \(h \to 0\), therefore we can make the following observation.</p>
<p>The gradient of the tangent to the curve \(y = f(x)\) at the point where \(x=a\) is 
\[
\lim\limits_{h \to 0} \frac{f(a+h)-f(a)}{h}.
\]
We will see this calculation frequently throughout the remainder of this week's material, so it is advisable to memorise it.</p>]]></text>
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