In Tricky Questions on the AP Calculus Exam we discussed the derivative of a function; recall that a function f(x) is differentiable exactly when the limit

exists for every x  in the domain of f(x).

## The Geometry of the Derivative

The derivative of a function f at a point x in the domain of f can be interpreted geometrically as the slope of the line tangent to the graph of f at that point. What does it mean for a line to be tangent to the graph of a function at a point? This has a more subtle answer. Pictorially, we can say the blue line in this graph is tangent to the graph of f at the point f(x):

It’s the line that “barely touches” the graph of f.

## Derivative Realtalk: The limit of secant lines

So it turns out that we express the slope of the blue line by taking the limit of the slopes of secant lines of f at point f(x). If you are wondering what the heck that is, the secant line of a function is just a line that goes through two specified points on the function. It can go through more than just those points, but it needs two points to be specified. Here is a diagram of secant lines  between f(a) and f(b), f(b) and f(c), f(c) and f(d), and f(d) and f(e), respectively:

Don’t let secants scare you. Line between two points on a function.

Now we want to now label in our original figure f(x) as f(a):

This is because we want to rewrite our derivative. It might not seem intuitive to rewrite this, but we want to be able to express the derivative in terms of the slope of a line, since the current version doesn’t look like that.

Remember that the general slope of a line has the form  So we want to try and do a little math to get there. We can clearly substitute a = x into the limit, because x does not vary:

Now we want to make the substitution a + h = x. Now it is clear that if   then  So substituting gives

This is a commonly used second definition of the derivative as the limit of the slopes of the secant lines through the points f(x) and f(a).

A rough sketch of this process, with the secants  through the point f(a) and  respectively with  would be:

So you can see that the secant lines are getting closer and closer to the tangent line, as

## Summary

You can think of the derivative of a function at a point as the slope of the line tangent to the function at that point. You can think of tangent lines as the limit of secant lines, and you can express differentiability by saying that the limit

Exists for all a in the domain of f.