1.4. THE DERIVATIVE FUNCTION
39
Now, recall the opening example of this section: we began with the function y =
f (x) = 4x − x 2 and used the limit definition of the derivative to show that f ′ (a) = 4 − 2a,
or equivalently that f ′ (x) = 4 − 2x. We subsequently graphed the functions f and f ′
as shown in Figure 1.18. Following Activity 1.10, we now understand that we could have
constructed a fairly accurate graph of f ′ (x) without knowing a formula for either f or f ′ .
At the same time, it is ideal to know a formula for the derivative function whenever it is
possible to find one.
In the next activity, we further explore the more algebraic approach to finding f ′ (x):
given a formula for y = f (x), the limit definition of the derivative will be used to develop
a formula for f ′ (x).
Activity 1.11.
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
f and x. For example, given a function p(z), we call its derivative p ′ (z).
(a) f (x) = 1
(b) g(t) = t
(c) p(z) = z 2
(d) q(s) = s 3
(e) F(t) =
1
t
(f) G(y) =
√
y
⊳
Summary
In this section, we encountered the following important ideas:
• The limit definition of the derivative, f ′ (x) = lim h→0
f (x+h)− f (x)
h
, produces a value
for each x at which the derivative is defined, and this leads to a new function whose
formula is y = f ′ (x). Hence we talk both about a given function f and its derivative
f ′ . It is especially important to note that taking the derivative is a process that starts
with a given function ( f ) and produces a new, related function ( f ′ ).
• There is essentially no difference between writing f ′ (a) (as we did regularly in Section 1.3) and writing f ′ (x). In either case, the variable is just a placeholder that is used
to define the rule for the derivative function.
39
Now, recall the opening example of this section: we began with the function y =
f (x) = 4x − x 2 and used the limit definition of the derivative to show that f ′ (a) = 4 − 2a,
or equivalently that f ′ (x) = 4 − 2x. We subsequently graphed the functions f and f ′
as shown in Figure 1.18. Following Activity 1.10, we now understand that we could have
constructed a fairly accurate graph of f ′ (x) without knowing a formula for either f or f ′ .
At the same time, it is ideal to know a formula for the derivative function whenever it is
possible to find one.
In the next activity, we further explore the more algebraic approach to finding f ′ (x):
given a formula for y = f (x), the limit definition of the derivative will be used to develop
a formula for f ′ (x).
Activity 1.11.
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
f and x. For example, given a function p(z), we call its derivative p ′ (z).
(a) f (x) = 1
(b) g(t) = t
(c) p(z) = z 2
(d) q(s) = s 3
(e) F(t) =
1
t
(f) G(y) =
√
y
⊳
Summary
In this section, we encountered the following important ideas:
• The limit definition of the derivative, f ′ (x) = lim h→0
f (x+h)− f (x)
h
, produces a value
for each x at which the derivative is defined, and this leads to a new function whose
formula is y = f ′ (x). Hence we talk both about a given function f and its derivative
f ′ . It is especially important to note that taking the derivative is a process that starts
with a given function ( f ) and produces a new, related function ( f ′ ).
• There is essentially no difference between writing f ′ (a) (as we did regularly in Section 1.3) and writing f ′ (x). In either case, the variable is just a placeholder that is used
to define the rule for the derivative function.