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Christopher Wirick

AP Calculus Vocabulary: The Derivative of a Function

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 cv-td_img3 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 cv-td_img6 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  cv-td_img8 then cv-td_img9 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 cv-td_img10b through the point f(a) and cv-td_img11 respectively with cv-td_img12 would be:


So you can see that the secant lines are getting closer and closer to the tangent line, as cv-td_img9



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.

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About Christopher Wirick

In addition to blogging about AP Calculus, Oakland resident Chris Wirick has played the oboe in Beijing and Berlin, studied math and cognitive science (the other CS) at Cal Berkeley, and can’t stop gardening, cooking and eating new foods.

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