Defining the Derivative of a Function at a Point

What the Derivative at a Point Means

Suppose you have a function and want to know how quickly the function is changing at one specific input, . You cannot determine an instantaneous rate of change simply by looking at one point. Instead, you examine what happens over smaller and smaller intervals around that point.

The derivative of at , written , measures the instantaneous rate of change of the function at that point. Geometrically, it is the slope of the tangent line to the graph at .

The key idea is that we begin with an average rate of change over an interval and then shrink that interval toward zero.

Average Rate of Change Comes First

Take two points on the graph of : and . The slope of the secant line connecting these two points is the average rate of change of the function over the interval.

This gives us the slope of a secant line, not yet the slope of a tangent line. To find the derivative at , we move the second point closer and closer to the first point.

The Difference Quotient

Instead of writing the second input as a completely separate value, write it as . Here, represents a small change in the input.

The average rate of change from to becomes:

Now comes the crucial step: let approach zero. As the second point gets arbitrarily close to , the secant line approaches the tangent line.

The Definition of the Derivative at a Point

The derivative of at is defined by the limit:

This is one of the most important definitions in all of AP Calculus. You should recognize that the derivative is not simply a formula you memorize. It comes directly from the idea of taking the limit of average rates of change.

There is an equivalent form of the definition. If the second point is represented by , then:

Both forms define exactly the same quantity.

What the Limit Is Doing

The expression represents the slope of a secant line between two points. When gets smaller, the second point moves closer to .

But there is an important subtlety: we do not simply substitute into the difference quotient. If we did, the denominator would become zero. Instead, we ask what value the quotient approaches as approaches zero.

The denominator is not supposed to become zero

The expression is evaluated through a limit. You simplify the difference quotient first, then evaluate the resulting limit. The fact that direct substitution initially produces is exactly why the limit is necessary.

Worked Examples

Example 1: Find a Derivative from the Definition

Use the limit definition of the derivative to find for .

Step 1: Start with the definition:

Step 2: Substitute the function:

Step 3: Expand the numerator:

Step 4: Cancel the common factor of :

Step 5: Take the limit:

Result:

Example 2: Derivative at a Specific Point

Use the definition of the derivative to find for .

Step 1: Use the point-specific definition:

Step 2: Calculate and :

Step 3: Substitute:

Step 4: Simplify the numerator:

Step 5: Cancel and take the limit:

Result:

Example 3: A Function with a Square Root

Use the limit definition to find for .

Step 1: Begin with the definition:

Step 2: Substitute the function:

Step 3: Rationalize the numerator by multiplying by the conjugate:

Step 4: Simplify:

Step 5: Take the limit:

Result:

Example 4: Connecting the Derivative to a Tangent Line

Let . Find the slope of the tangent line at , then write the equation of the tangent line.

Step 1: Use the derivative definition:

Step 2: Substitute:

Step 3: Simplify:

Step 4: Therefore the derivative at is:

Step 5: Find the corresponding point on the graph:

Result: Using point-slope form, the tangent line is .

Derivative Notation at a Point

AP Calculus uses several equivalent ways to represent the derivative. If , then the derivative at can be written as:

These expressions all refer to the same number: the instantaneous rate of change of when .

Derivative as a Function vs. Derivative at a Point

This distinction is extremely important. The expression represents the derivative function. It gives the slope at every input where the derivative exists.

By contrast, is one specific number. It is the value of the derivative function at one particular input.

Think of as the entire collection of instantaneous slopes, while is one member of that collection.

Common Mistakes

Confusing a secant slope with a tangent slope

The expression gives an average rate of change over an interval. The derivative requires taking the limit as the interval width approaches zero.

Plugging in zero before simplifying

If you substitute directly into the difference quotient, you may get . That does not mean the derivative does not exist. Simplify the quotient first, then evaluate the limit.

Forgetting that the derivative is a slope

At a point where the derivative exists, is the slope of the tangent line. It is also the instantaneous rate of change. These are two interpretations of the same number.

Using the wrong point in the formula

For , the point-specific definition is . Make sure the function value corresponds to the same input .

Practice Problems

Question 1

Use the limit definition of the derivative to find for .

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Question 2

Use the definition of the derivative to find for .

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Question 3

Use the limit definition to find the derivative of at .

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Question 4 (Conceptual)

Explain what represents geometrically if .

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