Section 1.4 The derivative function
Motivating Questions
How does the limit definition of the derivative of a function
lead to an entirely new (but related) functionWhat is the difference between writing
andHow is the graph of the derivative function
related to the graph ofWhat are some examples of functions
for which is not defined at one or more points?
Preview Activity 1.4.1.
Consider the function
Use the limit definition to compute the derivative values:
andObserve that the work to find
is the same, regardless of the value of Based on your work in (a), what do you conjecture is the value of How about (Note: you should not use the limit definition of the derivative to find either value.)Conjecture a formula for
that depends only on the value That is, in the same way that we have a formula for (recall ), see if you can use your work above to guess a formula for in terms of
Subsection 1.4.1 How the derivative is itself a function
In your work in Preview Activity 1.4.1 withDefinition 1.4.2.
Let
given a graph of
how does this graph lead to the graph of the derivative function andgiven a formula for
how does the limit definition of derivative generate a formula for
Activity 1.4.2.
For each given graph of
When you are finished with all 8 graphs, write several sentences that describe your overall process for sketching the graph of the derivative function, given the graph the original function. What are the values of the derivative function that you tend to identify first? What do you do thereafter? How do key traits of the graph of the derivative function exemplify properties of the graph of the original function?
Points where the slope of the tangent line is equal to zero are particularly important. Try finding these points first in your effort to plot
Activity 1.4.3.
For each of the listed functions, determine a formula for the derivative function. For the first two, determine the formula for the derivative by thinking about the nature of the given function and its slope at various points; do not use the limit definition. For the latter four, use the limit definition. Pay careful attention to the function names and independent variables. It is important to be comfortable with using letters other than
What is the slope of the function at every point?
What is the slope of the function at every point?
Subsection 1.4.2 Summary
The limit definition of the derivative,
produces a value for each at which the derivative is defined, and this leads to a new function It is especially important to note that taking the derivative is a process that starts with a given function ( ) and produces a new, related function ( ).There is essentially no difference between writing
(as we did regularly in Section 1.3) and writing In either case, the variable is just a placeholder that is used to define the rule for the derivative function.Given the graph of a function
we can sketch an approximate graph of its derivative by observing that heights on the derivative's graph correspond to slopes on the original function's graph.In Activity 1.4.2, we encountered some functions that had sharp corners on their graphs, such as the shifted absolute value function. At such points, the derivative fails to exist, and we say that
is not differentiable there. For now, it suffices to understand this as a consequence of the jump that must occur in the derivative function at a sharp corner on the graph of the original function.
Exercises 1.4.3 Exercises
1. The derivative function graphically.
Consider the function

(Note that you can click on the graph to get a larger version of it, and that it may be useful to print that larger version to be able to work with it by hand.)
Carefully sketch the derivative function of the given function (you will want to estimate values on the derivative function at different
at |
-3 | -1 | 1 | 3 |
the derivative is |
2. Applying the limit definition of the derivative.
Find a formula for the derivative of the function
(In the first answer blank, fill in the numerator of the difference quotient you use to evaluate the derivative. In the second, fill out the derivative you obtain after completing the limit calculation.)
3. Sketching the derivative.
For the function

Which of the following graphs is the derivative of
1
2
3
4
5
6
7
8
(Click on a graph to enlarge it.)
1.![]() |
2.![]() |
3.![]() |
4.![]() |
5.![]() |
6.![]() |
7.![]() |
8.![]() |
4. Comparing function and derivative values.
The graph of a function

At which of the labeled
x1
x2
x3
x4
x5
x6
x1
x2
x3
x4
x5
x6
x1
x2
x3
x4
x5
x6
x1
x2
x3
x4
x5
x6
5. Limit definition of the derivative for a rational function.
Let
Find
(i)
(ii)
(iii)
(iv)
6.
Let
On the axes provided at left in Figure 1.4.3, sketch a possible graph of
Explain why your graph meets the stated criteria.Conjecture a formula for the function
Use the limit definition of the derivative to determine the corresponding formula for Discuss both graphical and algebraic evidence for whether or not your conjecture is correct.
7.
Consider the function
Use the limit definition of the derivative to determine a formula for
Use a graphing utility to plot both
and your result for does your formula for generate the graph you expected?Use the limit definition of the derivative to find a formula for
whereCompare and contrast the formulas for
and you have found. How do the constants 5, 4, 12, and 3 affect the results?
8.
Let
Observe that for every value of
that satisfies the value of is constant. What does this tell you about the behavior of the graph of on this interval?On what intervals other than
do you expect to be a linear function? Why?At which values of
is not defined? What behavior does this lead you to expect to see in the graph ofSuppose that
On the axes provided at left in Figure 1.4.4, sketch an accurate graph of
9.
For each graph that provides an original function
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