From One Derivative to a Derivative Function
When you first learn derivatives, you often calculate the slope of a function at one particular point. For example, you might find the slope of at . But there is a more powerful way to think about the derivative: instead of finding one slope, we can create a new function that gives the slope at every point where the original function is differentiable.
If the original function is , its derivative is written . The new function tells us the instantaneous rate of change of at any input in its domain.
A derivative is not just a number. The derivative can itself be a function. Once you find that function, you can evaluate it at different values of to obtain the slope at different points.
The Definition of the Derivative Function
Start with the difference quotient. The average rate of change from to is:
To turn this into an instantaneous rate of change, we let the second point approach the first. If the horizontal distance between the two points is , then the second input is . Taking the limit as approaches zero gives the derivative function:
Notice what is different here. We are not substituting a particular value for . The variable remains , so the result is a function that can be evaluated at many different points.
Derivative at a Point vs. Derivative as a Function
These two ideas are closely related, but they are not identical.
The first expression describes the slope at every input where the derivative exists. The second gives one particular slope after evaluating the derivative function at .
Think of as a machine. Give it an input , and it returns the slope of at that input.
Worked Example 1: Finding the Derivative Function from the Definition
Find the derivative function of using the definition of the derivative.
Worked Example 2: Using the Derivative Function
Suppose . Find the slope of the graph at .
So the tangent line to the graph at has slope .
Worked Example 3: Derivative of a Polynomial
Find for , then determine the slope at .
The Derivative Function and the Original Graph
The derivative function contains information about the behavior of the original function. The most important connection is between the sign of the derivative and whether the original function is increasing or decreasing.
- If , the original function is increasing.
- If , the original function is decreasing.
- If , the tangent line is horizontal.
This is why the derivative function is so useful. Instead of examining every tangent line individually, we can study one function, , to understand how behaves.
What Does the Derivative Function Graph Look Like?
If you are given the graph of , the graph of represents the slopes of the tangent lines to the original graph.
For example, suppose the original function has a steep positive slope at some value of . Then the corresponding value of will be large and positive. If the original function has a steep negative slope, then will be large and negative.
The derivative graph is a graph of slopes. Each point on the derivative graph tells you the slope of the original function at the same input .
Worked Example 4: Reading a Derivative Function
Suppose the derivative function is .
Derivative Notation
AP Calculus uses several equivalent ways to write the derivative. You should be comfortable recognizing all of them.
The notation is particularly useful when emphasizing that the derivative is a function. The notation is especially useful when thinking about a derivative as a rate of change with respect to another variable.
Derivative as a Function vs. Derivative at a Point
A common AP Calculus distinction is whether a question asks you to find the derivative function or evaluate the derivative at a particular point.
In other words, finding the derivative function comes first. Evaluating that function at a particular input gives the derivative at that point.
Common Mistakes
The original function gives the output of the original relationship. The derivative function gives its instantaneous rate of change. They are different functions, even though they are built from the same original function.
The expression represents a function. It becomes one number only after you substitute a particular value of .
If a problem asks for the derivative function, keep as a variable. Substituting a particular value too early gives you a single derivative value rather than the complete derivative function.
Always connect the algebra back to meaning: represents the instantaneous rate of change of with respect to . In graphical problems, it represents the slope of the tangent line.
Practice Problems
Find the derivative function of .
Show answer
If , find the slope of the graph at .
Show answer
Use the definition of the derivative to find for .
Show answer
Suppose . What does tell you about the original function?
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon