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1.4. THE DERIVATIVE FUNCTION
1.4 The derivative function
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the limit definition of the derivative of a function f lead to an entirely
new (but related) function f ′ ?
• What is the difference between writing f ′ (a) and f ′ (x)?
• How is the graph of the derivative function f ′ (x) connected to the graph of f (x)?
• What are some examples of functions f for which f ′ is not defined at one or more
points?
Introduction
Given a function y = f (x), we now know that if we are interested in the instantaneous
rate of change of the function at x = a, or equivalently the slope of the tangent line to
y = f (x) at x = a, we can compute the value f ′ (a). In all of our examples to date, we
have arbitrarily identified a particular value of a as our point of interest: a = 1, a = 3, etc.
But it is not hard to imagine that we will often be interested in the derivative value for
more than just one a-value, and possibly for many of them. In this section, we explore
how we can move from computing simply f ′ (1) or f ′ (3) to working more generally with
f ′ (a), and indeed f ′ (x). Said differently, we will work toward understanding how the
so-called process of “taking the derivative” generates a new function that is derived from
the original function y = f (x). The following preview activity starts us down this path.
Preview Activity 1.4. Consider the function f (x) = 4x − x 2 .
(a) Use the limit definition to compute the following derivative values: f ′ (0), f ′ (1),
f ′ (2), and f ′ (3).
(b) Observe that the work to find f ′ (a) is the same, regardless of the value of a. Based
on your work in (a), what do you conjecture is the value of f ′ (4)? How about
f ′ (5)? (Note: you should not use the limit definition of the derivative to find either
value.)
(c) Conjecture a formula for f ′ (a) that depends only on the value a. That is, in the
same way that we have a formula for f (x) (recall f (x) = 4x − x 2 ), see if you can
use your work above to guess a formula for f ′ (a) in terms of a.
⊲⊳
1.4. THE DERIVATIVE FUNCTION
1.4 The derivative function
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How does the limit definition of the derivative of a function f lead to an entirely
new (but related) function f ′ ?
• What is the difference between writing f ′ (a) and f ′ (x)?
• How is the graph of the derivative function f ′ (x) connected to the graph of f (x)?
• What are some examples of functions f for which f ′ is not defined at one or more
points?
Introduction
Given a function y = f (x), we now know that if we are interested in the instantaneous
rate of change of the function at x = a, or equivalently the slope of the tangent line to
y = f (x) at x = a, we can compute the value f ′ (a). In all of our examples to date, we
have arbitrarily identified a particular value of a as our point of interest: a = 1, a = 3, etc.
But it is not hard to imagine that we will often be interested in the derivative value for
more than just one a-value, and possibly for many of them. In this section, we explore
how we can move from computing simply f ′ (1) or f ′ (3) to working more generally with
f ′ (a), and indeed f ′ (x). Said differently, we will work toward understanding how the
so-called process of “taking the derivative” generates a new function that is derived from
the original function y = f (x). The following preview activity starts us down this path.
Preview Activity 1.4. Consider the function f (x) = 4x − x 2 .
(a) Use the limit definition to compute the following derivative values: f ′ (0), f ′ (1),
f ′ (2), and f ′ (3).
(b) Observe that the work to find f ′ (a) is the same, regardless of the value of a. Based
on your work in (a), what do you conjecture is the value of f ′ (4)? How about
f ′ (5)? (Note: you should not use the limit definition of the derivative to find either
value.)
(c) Conjecture a formula for f ′ (a) that depends only on the value a. That is, in the
same way that we have a formula for f (x) (recall f (x) = 4x − x 2 ), see if you can
use your work above to guess a formula for f ′ (a) in terms of a.
⊲⊳