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Tangent Lines and
Definition of the Derivative

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Suppose we are given a function  and a point in the domain of  . How can we find the tangent line  to the graph of  at the point (   , ) on the graph?   Here is a picture of the sort of thing we are after.

The blue curve is the graph of the function  , where , and the red line is the tangent line to the graph of at .   You can see on the picture what is meant by a tangent line:  it is the line which intersects the graph of  at the given point, and is "parallel" to the graph at that point.  

It is important to realize that this is the only information we have about the tangent  line.  In particular, we do not know whether the tangent line will intersect the graph of a second time.

Take as an example.  The blue curve is the graph of g, and the red line is the tangent line to the graph of g at . Observe that the red line intersect the graph of g at two points other than .

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The tangent line problem then is to find the equation of a line when we only know one point on the line.  Of course, if we also knew the slope of the line, we could write down its equation in point-slope form, so the problem becomes: how do we find the slope of the tangent line?

If we are given two  points on the graph of  , we can easily find the equation of the line joining them.  ( A line joining two points on a graph is called a secant line. )

 

In the first example, , we chose x=3 as the second point.  Let's examine what happens as this point is chosen closer and closer to the starting point x=1.

The following commands create the various components of the animation.

 
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You can see that as the moving point approaches the fixed one (x = 1), the secant line swings round and approaches the tangent line.

Presumably, therefore, its slope approaches the slope of the tangent line.

We say that the slope of the tangent line is the limit of the slope of the secant line. In symbols, the slope of the secant line through the points 

( , ) and ( , )  is

 

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and the slope of the tangent line at (  , ) equals to

   

The following table shows how changes as is closer and closer to the .

	1/2	3/4	5/6	7/8	9/10	11/12	13/14	15/16	17/18	19/20
	1.5	1.75	1.8333	1.875	1.9	1.9167	1.9286	1.9375	1.9444	1.95

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