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Any two points can be used in determining the slope of a line, because the slope is constant throughout. Now consider the challenge of trying to find the slope of the following figure:

Figure %: f (x) = x3 +4x2 - 3

It should be readily apparent that there is no single slope for this figure. Instead, the curve has a different slope at each separate point. Therefore, for non-linear figures, it makes sense only to talk about the slope at a particular point.

Example: Find the slope of the graph of f at an arbitrary point x.

To visualize what needs to be done, let's consider an arbitrary function f and delineate an arbitrary point x:

Figure %: An arbitrary point x on a function f

The question asks us to find the slope of f at this arbitrary point x. The method with which we are already familiar calls for picking two points on the curve and calculating , so lets proceed this way first. Clearly, one of the points we should use is the point (x, f (x)), since this is the point on the graph where we want to find the slope. But what should be chosen as the other point? Intuitively, it might seem that no other point would yield the right answer, since we are interested in the slope at the single point (x, f (x)) only. Nevertheless, let us pick an arbitrary point h units away on the x-axis, (x + h, f (x + h)):

Figure %: The nearby point (x + h, f (x + h))

Now, we can calculate the quantity for these two points:


 =  
 =  

This quantity,

   

is called the difference quotient. It does not represent the slope of the graph at (x, f (x)). Rather, it represents the slope of the secant line that goes through the points (x, f (x)) and (x + h, f (x + h)):

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