56
1.6. THE SECOND DERIVATIVE
Because f ′ is itself a function, it is perfectly feasible for us to consider the derivative of
the derivative, which is the new function y = [ f ′ (x)] ′ . We call this resulting function the
second derivative of y = f (x), and denote the second derivative by y = f ′′ (x). Due to the
presence of multiple possible derivatives, we will sometimes call f ′ “the first derivative” of
f , rather than simply “the derivative” of f . Formally, the second derivative is defined by
the limit definition of the derivative of the first derivative:
f
′′ (x) = lim
h→0
f ′ (x + h) − f ′ (x)
h
.
We note that all of the established meaning of the derivative function still holds, so
when we compute y = f ′′ (x), this new function measures slopes of tangent lines to the
curve y = f ′ (x), as well as the instantaneous rate of change of y = f ′ (x). In other words,
just as the first derivative measures the rate at which the original function changes, the
second derivative measures the rate at which the first derivative changes. This means that
the second derivative tracks the instantaneous rate of change of the instantaneous rate
of change of f . That is, the second derivative will help us to understand how the rate of
change of the original function is itself changing.
Concavity
In addition to asking whether a function is increasing or decreasing, it is also natural
to inquire how a function is increasing or decreasing. To begin, there are three basic
behaviors that an increasing function can demonstrate on an interval, as pictured in
Figure 1.29: the function can increase more and more rapidly, increase at the same rate, or
increase in a way that is slowing down. Fundamentally, we are beginning to think about
how a particular curve bends, with the natural comparison being made to lines, which
don’t bend at all. More than this, we want to understand how the bend in a function’s
graph is tied to behavior characterized by the first derivative of the function.
For the leftmost curve in Figure 1.29, picture a sequence of tangent lines to the curve.
As we move from left to right, the slopes of those tangent lines will increase. Therefore,
the rate of change of the pictured function is increasing, and this explains why we say this
function is increasing at an increasing rate. For the rightmost graph in Figure 1.29, observe
that as x increases, the function increases but the slope of the tangent line decreases,
hence this function is increasing at a decreasing rate.
Of course, similar options hold for how a function can decrease. Here we must be
extra careful with our language, since decreasing functions involve negative slopes, and
negative numbers present an interesting situation in the tension between common language
and mathematical language. For example, it can be tempting to say that “−100 is bigger
than −2.” But we must remember that when we say one number is greater than another,
this describes how the numbers lie on a number line: x < y provided that x lies to the
left of y. So of course, −100 is less than −2. Informally, it might be helpful to say that
1.6. THE SECOND DERIVATIVE
Because f ′ is itself a function, it is perfectly feasible for us to consider the derivative of
the derivative, which is the new function y = [ f ′ (x)] ′ . We call this resulting function the
second derivative of y = f (x), and denote the second derivative by y = f ′′ (x). Due to the
presence of multiple possible derivatives, we will sometimes call f ′ “the first derivative” of
f , rather than simply “the derivative” of f . Formally, the second derivative is defined by
the limit definition of the derivative of the first derivative:
f
′′ (x) = lim
h→0
f ′ (x + h) − f ′ (x)
h
.
We note that all of the established meaning of the derivative function still holds, so
when we compute y = f ′′ (x), this new function measures slopes of tangent lines to the
curve y = f ′ (x), as well as the instantaneous rate of change of y = f ′ (x). In other words,
just as the first derivative measures the rate at which the original function changes, the
second derivative measures the rate at which the first derivative changes. This means that
the second derivative tracks the instantaneous rate of change of the instantaneous rate
of change of f . That is, the second derivative will help us to understand how the rate of
change of the original function is itself changing.
Concavity
In addition to asking whether a function is increasing or decreasing, it is also natural
to inquire how a function is increasing or decreasing. To begin, there are three basic
behaviors that an increasing function can demonstrate on an interval, as pictured in
Figure 1.29: the function can increase more and more rapidly, increase at the same rate, or
increase in a way that is slowing down. Fundamentally, we are beginning to think about
how a particular curve bends, with the natural comparison being made to lines, which
don’t bend at all. More than this, we want to understand how the bend in a function’s
graph is tied to behavior characterized by the first derivative of the function.
For the leftmost curve in Figure 1.29, picture a sequence of tangent lines to the curve.
As we move from left to right, the slopes of those tangent lines will increase. Therefore,
the rate of change of the pictured function is increasing, and this explains why we say this
function is increasing at an increasing rate. For the rightmost graph in Figure 1.29, observe
that as x increases, the function increases but the slope of the tangent line decreases,
hence this function is increasing at a decreasing rate.
Of course, similar options hold for how a function can decrease. Here we must be
extra careful with our language, since decreasing functions involve negative slopes, and
negative numbers present an interesting situation in the tension between common language
and mathematical language. For example, it can be tempting to say that “−100 is bigger
than −2.” But we must remember that when we say one number is greater than another,
this describes how the numbers lie on a number line: x < y provided that x lies to the
left of y. So of course, −100 is less than −2. Informally, it might be helpful to say that