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.
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).
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.