1.4. THE DERIVATIVE FUNCTION
35
How the derivative is itself a function
In your work in Preview Activity 1.4 with f (x) = 4x − x 2 , you may have found several
patterns. One comes from observing that f ′ (0) = 4, f ′ (1) = 2, f ′ (2) = 0, and f ′ (3) = −2.
That sequence of values leads us naturally to conjecture that f ′ (4) = −4 and f ′ (5) = −6.
Even more than these individual numbers, if we consider the role of 0, 1, 2, and 3 in the
process of computing the value of the derivative through the limit definition, we observe
that the particular number has very little effect on our work. To see this more clearly,
we compute f ′ (a), where a represents a number to be named later. Following the now
standard process of using the limit definition of the derivative,
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
= lim
h→0
4(a + h) − (a + h) 2 − (4a − a 2 )
h
= lim
h→0
4a + 4h − a 2 − 2ha − h 2 − 4a + a 2
h
= lim
h→0
4h − 2ha − h 2
h
= lim
h→0
h(4 − 2a − h)
h
= lim
h→0
(4 − 2a − h).
Here we observe that neither 4 nor 2a depend on the value of h, so as h → 0, (4−2a−h) →
(4 − 2a). Thus, f ′ (a) = 4 − 2a.
This observation is consistent with the specific values we found above: e.g., f ′ (3) =
4 − 2(3) = −2. And indeed, our work with a confirms that while the particular value of a
at which we evaluate the derivative affects the value of the derivative, that value has almost
no bearing on the process of computing the derivative. We note further that the letter
being used is immaterial: whether we call it a, x, or anything else, the derivative at a given
value is simply given by “4 minus 2 times the value.” We choose to use x for consistency
with the original function given by y = f (x), as well as for the purpose of graphing the
derivative function, and thus we have found that for the function f (x) = 4x − x 2 , it follows
that f ′ (x) = 4 − 2x.
Because the value of the derivative function is so closely linked to the graphical
behavior of the original function, it makes sense to look at both of these functions plotted
on the same domain. In Figure 1.18, on the left we show a plot of f (x) = 4x − x 2 together
with a selection of tangent lines at the points we’ve considered above. On the right, we
show a plot of f ′ (x) = 4 − 2x with emphasis on the heights of the derivative graph at the
same selection of points. Notice the connection between colors in the left and right graph:
the green tangent line on the original graph is tied to the green point on the right graph
in the following way: the slope of the tangent line at a point on the lefthand graph is the
35
How the derivative is itself a function
In your work in Preview Activity 1.4 with f (x) = 4x − x 2 , you may have found several
patterns. One comes from observing that f ′ (0) = 4, f ′ (1) = 2, f ′ (2) = 0, and f ′ (3) = −2.
That sequence of values leads us naturally to conjecture that f ′ (4) = −4 and f ′ (5) = −6.
Even more than these individual numbers, if we consider the role of 0, 1, 2, and 3 in the
process of computing the value of the derivative through the limit definition, we observe
that the particular number has very little effect on our work. To see this more clearly,
we compute f ′ (a), where a represents a number to be named later. Following the now
standard process of using the limit definition of the derivative,
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
= lim
h→0
4(a + h) − (a + h) 2 − (4a − a 2 )
h
= lim
h→0
4a + 4h − a 2 − 2ha − h 2 − 4a + a 2
h
= lim
h→0
4h − 2ha − h 2
h
= lim
h→0
h(4 − 2a − h)
h
= lim
h→0
(4 − 2a − h).
Here we observe that neither 4 nor 2a depend on the value of h, so as h → 0, (4−2a−h) →
(4 − 2a). Thus, f ′ (a) = 4 − 2a.
This observation is consistent with the specific values we found above: e.g., f ′ (3) =
4 − 2(3) = −2. And indeed, our work with a confirms that while the particular value of a
at which we evaluate the derivative affects the value of the derivative, that value has almost
no bearing on the process of computing the derivative. We note further that the letter
being used is immaterial: whether we call it a, x, or anything else, the derivative at a given
value is simply given by “4 minus 2 times the value.” We choose to use x for consistency
with the original function given by y = f (x), as well as for the purpose of graphing the
derivative function, and thus we have found that for the function f (x) = 4x − x 2 , it follows
that f ′ (x) = 4 − 2x.
Because the value of the derivative function is so closely linked to the graphical
behavior of the original function, it makes sense to look at both of these functions plotted
on the same domain. In Figure 1.18, on the left we show a plot of f (x) = 4x − x 2 together
with a selection of tangent lines at the points we’ve considered above. On the right, we
show a plot of f ′ (x) = 4 − 2x with emphasis on the heights of the derivative graph at the
same selection of points. Notice the connection between colors in the left and right graph:
the green tangent line on the original graph is tied to the green point on the right graph
in the following way: the slope of the tangent line at a point on the lefthand graph is the