36
1.4. THE DERIVATIVE FUNCTION
1
2
3
4
-4
-3
-2
-1
1
2
3
4
m = −4
y = f (x)
m = 4
m = 2
m = 0
m = −2
1
2
3
4
-4
-3
-2
-1
1
2
3
4
y = f ′ (x)
(0, 4)
(1, 2)
(2, 0)
(3, −2)
(4, −4)
Figure 1.18: The graphs of f (x) = 4x − x 2 (at left) and f ′ (x) = 4 − 2x (at right). Slopes on
the graph of f correspond to heights on the graph of f ′ .
same as the height at the corresponding point on the righthand graph. That is, at each
respective value of x, the slope of the tangent line to the original function at that x-value
is the same as the height of the derivative function at that x-value. Do note, however, that
the units on the vertical axes are different: in the left graph, the vertical units are simply
the output units of f . On the righthand graph of y = f ′ (x), the units on the vertical axis
are units of f per unit of x.
Of course, this relationship between the graph of a function y = f (x) and its derivative
is a dynamic one. An excellent way to explore how the graph of f (x) generates the graph
of f ′ (x) is through a java applet. See, for instance, the applets at http://gvsu.edu/s/5C
or http://gvsu.edu/s/5D, via the sites of Austin and Renault 5 .
In Section 1.3 when we first defined the derivative, we wrote the definition in terms of
a value a to find f ′ (a). As we have seen above, the letter a is merely a placeholder, and it
often makes more sense to use x instead. For the record, here we restate the definition of
the derivative.
Definition 1.4. Let f be a function and x a value in the function’s domain. We define
the derivative of f with respect to x at the value x, denoted f ′ (x), by the formula f
′ (x) =
lim
h→0
f (x + h) − f (x)
h
, provided this limit exists.
5 David Austin, http://gvsu.edu/s/5r; Marc Renault, http://gvsu.edu/s/5p.
1.4. THE DERIVATIVE FUNCTION
1
2
3
4
-4
-3
-2
-1
1
2
3
4
m = −4
y = f (x)
m = 4
m = 2
m = 0
m = −2
1
2
3
4
-4
-3
-2
-1
1
2
3
4
y = f ′ (x)
(0, 4)
(1, 2)
(2, 0)
(3, −2)
(4, −4)
Figure 1.18: The graphs of f (x) = 4x − x 2 (at left) and f ′ (x) = 4 − 2x (at right). Slopes on
the graph of f correspond to heights on the graph of f ′ .
same as the height at the corresponding point on the righthand graph. That is, at each
respective value of x, the slope of the tangent line to the original function at that x-value
is the same as the height of the derivative function at that x-value. Do note, however, that
the units on the vertical axes are different: in the left graph, the vertical units are simply
the output units of f . On the righthand graph of y = f ′ (x), the units on the vertical axis
are units of f per unit of x.
Of course, this relationship between the graph of a function y = f (x) and its derivative
is a dynamic one. An excellent way to explore how the graph of f (x) generates the graph
of f ′ (x) is through a java applet. See, for instance, the applets at http://gvsu.edu/s/5C
or http://gvsu.edu/s/5D, via the sites of Austin and Renault 5 .
In Section 1.3 when we first defined the derivative, we wrote the definition in terms of
a value a to find f ′ (a). As we have seen above, the letter a is merely a placeholder, and it
often makes more sense to use x instead. For the record, here we restate the definition of
the derivative.
Definition 1.4. Let f be a function and x a value in the function’s domain. We define
the derivative of f with respect to x at the value x, denoted f ′ (x), by the formula f
′ (x) =
lim
h→0
f (x + h) − f (x)
h
, provided this limit exists.
5 David Austin, http://gvsu.edu/s/5r; Marc Renault, http://gvsu.edu/s/5p.