1.6. HIGHER-ORDER DERIVATIVES
1.6
125
Higher-Order Derivatives
We have defined the instantaneous velocity and acceleration of an object: If p = p(t) is the
position of an object at time t, then the velocity v = v(t) is equal to the derivative p0 (t), and
the acceleration a = a(t) is equal to the derivative v 0 (t).
This means that a(t) is the derivative, with respect to t, of the derivative, with respect to
t, of the position p(t). Such iterated derivatives occur often enough and are important enough
that we adopt special terminology and notation.
Definition 1.6.1. Suppose that we have a function y = f (x). Then, the derivative of f 0 (x),
with respect to x, is called the second derivative of f , with respect to x. This is written
as
d2 y
d2 f
d2
f 00 (x),
(f (x))00 ,
y 00 ,
,
,
or
(f (x)).
dx2
dx2
dx2
The third derivative of f , with respect to x, is defined to be the derivative of f 00 (x),
with respect to x.
More generally, for any positive integer n, the n-th derivative of f , with respect to x
is obtained by iterating the derivative, with respect to x, n times, i.e., the n-th derivative of
f , with respect to x, is the derivative, with respect to x, of the (n − 1)-st derivative of f ,
with respect to x. This is written as
f (n) (x),
(f (x))(n) ,
y (n) ,
dn y
,
dxn
dn f
, or
dxn
The integer n is called the order of the derivative.
If n ≥ 2, f (n) (x) is referred to as a higher-order derivative.
dn
(f (x)).
dxn
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CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE
Remark 1.6.2. It is sometimes convenient to refer to f itself as the 0-th derivative of f , i.e.,
f (0) (x) is simply f (x).
It is a matter of personal taste at what order you stop using the prime notation. It is not
uncommon to see f 000 (x) for the third derivative of f , with respect to x. Beyond the third
derivative, it is usual to start superscripting by the order number in parentheses, as we wrote
above, e.g., to write f (4) (x) for the fourth derivative.
dn y
may seem strange. They appear where they do
dxn
because they are meant to indicate the application, n times, of the differentiation with respect
d
to x operation,
. For instance,
dx
The placement of the order indicators in
d
dx
d
(f (x))
dx
=
d
dx
2
(f (x)) =
d2
d2 f
(f
(x))
=
.
dx2
dx2
Considering acceleration as the second derivative of position, with respect to time, is one of
the most standard uses for higher-order derivatives. Actually, the third derivative of position,
with respect to time, also has physical significance.
Recall Newton’s 2nd Law of Motion from Example 1.4.12: if the mass m of an object is
constant, then the net force, F , acting on the object is equal to ma, where a is the acceleration
of the object. This means that the instantaneous rate of change of the net force on the object,
with respect to time t, is given by
da
dF
=m .
dt
dt
In other words, the rate of change of the net force on the object is proportional to the rate
of change of the acceleration. When you experience a “jerk”, what you feel is a change in the
net force acting on you; that is, dF/dt measures how much of a jerk you feel. Since dF/dt is
proportional to da/dt, da/dt also measures how much of a jerk you feel.
This motivates the following definition.
Definition 1.6.3. The derivative of acceleration, with respect to time, is called jerk. Thus,
jerk is the third derivative of position, with respect to time.
1.6. HIGHER-ORDER DERIVATIVES
127
Remark 1.6.4. You should keep in mind the physical units on the higher-order derivatives.
We know that the derivative f 0 (x) of a function, f , with respect to a variable/quantity x, has
units given by the units of f divided by the units of x. For instance, if p = p(t) is the position,
in meters, of an object at time t seconds, then the units of p0 (t) are m/s, meters per second.
What about the higher-order derivatives? The second derivatives f 00 (x) or p00 (t) are the
derivatives, with respect to x and t, respectively, of the derivatives. Hence, the units of the
second derivative are the units of the derivative of the original function divided again by the
units of x or t, respectively. This means that the units of the second derivative are: the units
of the original function divided by the units of the variable, divided a second time by the units
of the variable. For instance, the units of acceleration a = p00 (t) are (m/s)/s, meters per second
per second, or m/s2 , meters per second squared.
By iterating this, we find that the units of f (n) (x) are the units of f divided by the units of
x raised to the n-th power.
You may be wondering if there is a graphical way of “seeing” the higher-order derivatives
or, at least, the second derivative. The answer is “yes”, and we shall discuss this at length in
Section 3.2, but we can have a preliminary discussion now.
The second derivative of f , with respect to x, is the instantaneous rate of change of the
instantaneous rate of change of f , with respect to x. But, graphically, the instantaneous rate of
change of f , with respect to x, is the slope of the tangent line to the graph of y = f (x). Thus,
the second derivative is the rate of change of the slope of the tangent lines to the graph of f ,
and it is relatively easy to see on a graph where this is positive or negative.
Example 1.6.5.
Consider the graph in Figure 1.31, where we have cycled the colors blue, green, and red to
help distinguish pieces of tangent lines (or you can click on the picture to view an animation).
What do you see in the graph? When x = −2, the slope of the tangent line is large. As
you let x increase to −1, the slopes of the tangent lines decrease, i.e., the slopes get smaller.
Therefore, the rate of change of the slope of the tangent lines is negative between x = −2 and
x = −1. In other words, the second derivative is negative.
After x = −1 as x increase to x = 0.5, the slope of the tangent line becomes negative, and
becomes even more negative as we closer to x = 0.5. Therefore, the second derivative is still
negative until around x = 0.5.
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CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE
7.5
5
2.5
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
-2.5
-5
-7.5
Figure 1.31: Tangent lines to a cubic graph.
Between x = 0.5 and x = 2, the slopes go from being pretty negative to being close to zero,
i.e., the slopes are increasing, and so the rate of change of the slopes is positive. This means
that the second derivative is positive between x = 0.5 and x = 2.
For x > 2, it appears that the slopes of the tangent lines continue to increase as x increases.
Hence, the second derivative should be positive for these x values as well.
If you look at this example long enough, you should see that the second derivative, f 00 (x), is
negative where the graph of f is concave down, i.e., where the graph curves downward. What
you’re seeing when you observe that the graph is concave down is precisely that the slopes of
the tangent lines are decreasing.
The graph is concave up, i.e., curves upward, where f 00 (x) is positive; exactly what you see
is the slopes of the tangent lines increasing.
In Section 2.1, we will discuss how to quickly calculate derivatives of polynomial functions.
Then we shall return to this cubic graph in Example 2.1.11.
Let us record our observations above in a definition.
1.6. HIGHER-ORDER DERIVATIVES
129
Definition 1.6.6. We say that the graph of f is concave up on an open interval I if and
only if f 00 (x) exists and is positive for all x in I.
The graph of f is concave down on an open interval I if and only if f 00 (x) exists and
is negative for all x in I.
If the concavity of the graph of f switches at x = b, that is, if b is in the domain of
f , and the graph of f is concave up on some open interval (a, b) and concave down on
some open interval (b, c), or vice-versa, then the point b (or the point (b, f (b))) is called an
inflection point of the graph of f .
Note that, if the second derivative of f exists on an open interval around b, and b is an
inflection point of the graph of f , then f 00 (b) must be zero. However, it is not true that every
place where f 00 = 0 must be an inflection point.
An easy example is provided by f (x) = x, for which f 0 (x) = 1 and f 00 (x) = 0. The graph
of y = x is, of course, a straight line, and so is not concave up or down anywhere; hence, the
concavity never switches signs, and there are no inflection points. To give a more interesting
example, we have to look ahead and assume Theorem 2.1.1. Then it is easy for us to show that,
if g(x) = x4 , then g 00 (x) = 12x2 . Thus, g 00 (0) = 0, but the graph of g is concave up for all x < 0
and for all x > 0, i.e., the concavity does not switch at x = 0, so x = 0 is not an inflection point
of the graph of g.
Example 1.6.7.
Suppose that the position of an object at time t hours is p = p(t) = 5t2 − 2t − 1 miles.
The graph of this function appears in Figure 1.32. This graph is concave up everywhere (or,
at least, in the portion shown), and so, from our discussion in the previous example, we expect
to find that the second derivative is positive for all t. We shall calculate the second derivative,
with respect to time, that is, we shall calculate the acceleration a and see that it is, in fact,
always positive.
Using Proposition 1.4.13, Proposition 1.4.11, and Proposition 1.4.5, we quickly calculate the
velocity v (mixing derivative notations, to continue helping you to adjust to using the various
notations interchangeably):
v =
dp
= (5t2 − 2t − 1)0 = (5t2 )0 + (−2t)0 + (−1)0 =
dt
5(t2 )0 − 2(t)0 + 0 = 5(2t) − 2 · 1 = 10t − 2 mph.
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CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE
3
p
2
1
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1
-2
-3
Figure 1.32: Position as a function of time.
Now, we find
a =
dv
d2 p
=
= (10t − 2)0 = 10 mph/hr.
dt
dt2
Therefore, the acceleration is constant and, as we expected from the graph, always positive.
The calculation of the jerk is easy:
j =
d3 p
da
=
= (10)0 = 0 (mph/hr)/hr.
dt
dt3
Remark 1.6.8. While we do not yet have rules for differentiation that let us look at more
complex examples of second derivatives in an easy manner, we should mention that second
derivatives are talked about fairly often in the news.
When it is reported that “the rate of increase in unemployment is decreasing” or “the rate
of increase in the average temperature of the Earth is increasing”, these are references to the
second derivative.
Think about it. If U (t) is the number of unemployed people (belonging to some group of
possible workers), then dU/dt is the (instantaneous) rate of change of U , with respect to t; when
this rate is positive, it is frequently called the rate of increase in unemployment. To say that
the rate of increase in unemployment is decreasing is to say that dU/dt is decreasing, i.e., the
rate of change of dU/dt, with respect to time, is negative. Therefore, the phrase “the rate of
1.6. HIGHER-ORDER DERIVATIVES
131
increase in unemployment is decreasing” tells us implicitly that dU/dt is positive, and explicitly
that d2 U/dt2 is negative.
We conclude this section with a method for using second derivatives to detect local extrema
and, in special cases, global extrema.
Theorem 1.6.9. (The Second Derivative Test for Extrema) Suppose that f 0 (c) = 0.
Then,
1. if f 00 (c) > 0, then f attains a local minimum value at c;
2. if f 00 (c) < 0, then f attains a local maximum value at c.
Furthermore, suppose that f is continuous on an interval I, differentiable on the interior of
I, and that c is the only critical point of f in the interior of I. Then,
a. if f 00 (c) > 0, then f attains a global minimum value at c;
b. if f 00 (c) < 0, then f attains a global maximum value at c.
Proof. We first show (1) and (2).
Assume f 0 (c) = 0 and f 00 (c) > 0. Note that the existence of f 00 (c) implies that f 0 exists on
some open interval around c, which also implies that f is continuous on an interval around c.
We have
f 00 (c) = lim
h→0
f 0 (c + h)
f 0 (c + h) − f 0 (c)
= lim
> 0.
h→0
h
h
Looking at the two one-sided limits as h → 0+ and h → 0− , we see that there exists an open
interval (c, b) on which f 0 must be positive, and an open interval (a, c) on which f 0 must be
negative. Thus, we may apply the First Derivative Test, Theorem 1.5.20, to conclude that f
attains a local minimum value at c.
The local maximum proof is completely analogous, and we leave it as an exercise.
We now show (a) and (b).
Now suppose that f is continuous on an interval I, differentiable on the interior of I, and
that c is the only critical point of f in the interior of I. We wish to apply the First Derivative
Test again.
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CHAPTER 1. RATES OF CHANGE AND THE DERIVATIVE
Let (a, b) be the interior of I. Then, the sign of f 0 cannot change on the interval (a, c) or
on the interval (c, b), for if the sign did change, then, by the Intermediate Value Theorem for
Derivatives, Theorem 1.A.25, we would have to have f 0 (x) = 0 for some x in (a, b) other than
x = c; a contradiction of our assumption. Now, either the conclusion or the proof of the local
extrema cases of this theorem tells us that the sign of f 0 actually switches at c. Thus, we apply
the First Derivative Test to finish the proof.
Remark 1.6.10. Note that the Second Derivative test says nothing about what happens if
f 00 (c) = 0. There is a good reason for this; if f 0 (c) = 0 and f 00 (c) = 0, then f may attain a local
maximum value, a local minimum value, or neither at c.
Assuming that enough derivatives of f exist to finally obtain one that is non-zero, the
question is whether that first non-zero derivative f (n) (c) 6= 0 occurs when n is even or n is odd.
If this n is odd, then f does not attain a local extreme value at c. If this n is even, then
the situation is precisely like that of the Second Derivative Test: if f (n) (c) > 0, then f attains a
local minimum value at c; if f (n) (c) < 0, then f attains a local maximum value at c. We leave
the proof of this as an exercise.
A final comment on the Second Derivative Test: while the Second Derivative Test does
not require the continuity of f 00 on an open interval around c, this would, nonetheless, be the
standard case. Assuming this to be true, it is easy to picture the conclusion of the Second
Derivative Test, since we know that f 00 being positive on an interval means that the graph of f
is concave up on the interval, and f 00 being negative on an interval means that the graph of f
is concave down on the interval.
f ´(c) = 0.
f´´(c) < 0.
f ´(c) = 0.
f´´(c) > 0.
Figure 1.33: A typical local minimum.
Figure 1.34: A typical local maximum.