AP Calculus (AB)
The Definition of the Derivative at a Point

A classic AP exam question type centers on the student knowing ... and knowing well ... how to recognise the definition of the derivative at a point. We spend so much time using the various "tricks" and formulas to calculate derivatives that we often overlook the fact that derivatives are rooted in limits. There are three common ways to represent the derivative of a function as a limit.

Perhaps the most common representation is

This form gives the "formula" for the derivative at any x-value. However, you should also be familiar with the following two methods to find the derivative of a function at a point, x = c:

and

A "worked" example of each should show how to recognize each in different situations. For clarity, the method involving h will be called "as h approaches 0" and the other method will be called "as x approaches c."

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The limit at a point as "as h approaches 0"

Find the derivative of the function as x approaches 2 using the definition of the derivative at a point:

First, let us look at the graph of the function and the behavior in the neighborhood when x = 2. We see that the function is clearly decreasing at the point in question, thus we expect the derivative to be negative.

The numerator of the derivative formula asks for the following:

f (c + h) and f (c)

If we evaluate these separately and then insert our results into the limit expression, we should have greater success than if we tried to do everything at once. Thus, ...

and

This yields the following limit calculation:

Thus, the derivative of the function g(x) at the point on the graph (2, 1) is –1. Could this derivative have been found with other differentiation "tricks" (here with either the quotient rule or the chain rule)? Certainly!

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The limit at a point as "as x approaches c"

Find the derivative of the function as x approaches 3 using the definition of the derivative at a point as x approaches c :

Here we see that . Thus, ...

Therefore, . This is the derivative at the point (when x = 3). This is confirmed with technology, evaluating the derivative at 3, as shown by the graphing calculator. Again the question: Could this derivative have been found with other differentiation "tricks" (namely, the chain rule)? Again, the answer is yes.

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AP Style Questions Involving the Derivative at a Point:

The AP exam often formats questions involving the derivative at a point as "simple" limit questions, with no mention of the derivative. Examples include the following (note, these are examples, not specific AP questions):

Evaluate either of the following limits:

or

      

Here the successful student will recognize each of these as the definition of the derivative at a point. In this case, the function is y = sin x and the point in question is where x = p/6.

Thus, rather than attempting to evaluate a "messy" limit, turn the question into a simple derivative problem and evaluate the derivative at the indicated x-value. In this case,

and we know

thus

.

Without realizing this is a "derivative at a point" question, you may lose valuable time attempting to solve a limit problem when the solution can be found in seconds. Thus, the key to success on questions of this sort is your familiarity with the definition of derivatives at points.

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