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1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
1.3 The derivative of a function at a point
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How is the average rate of change of a function on a given interval defined, and
what does this quantity measure?
• How is the instantaneous rate of change of a function at a particular point defined?
How is the instantaneous rate of change linked to average rate of change?
• What is the derivative of a function at a given point? What does this derivative
value measure? How do we interpret the derivative value graphically?
• How are limits used formally in the computation of derivatives?
Introduction
An idea that sits at the foundations of calculus is the instantaneous rate of change of a
function. This rate of change is always considered with respect to change in the input
variable, often at a particular fixed input value. This is a generalization of the notion
of instantaneous velocity and essentially allows us to consider the question “how do we
measure how fast a particular function is changing at a given point?” When the original
function represents the position of a moving object, this instantaneous rate of change is
precisely velocity, and might be measured in units such as feet per second. But in other
contexts, instantaneous rate of change could measure the number of cells added to a
bacteria culture per day, the number of additional gallons of gasoline consumed by going
one mile per additional mile per hour in a car’s velocity, or the number of dollars added
to a mortgage payment for each percentage increase in interest rate. Regardless of the
presence of a physical or practical interpretation of a function, the instantaneous rate of
change may also be interpreted geometrically in connection to the function’s graph, and
this connection is also foundational to many of the main ideas in calculus.
In what follows, we will introduce terminology and notation that makes it easier
to talk about the instantaneous rate of change of a function at a point. In addition,
just as instantaneous velocity is defined in terms of average velocity, the more general
instantaneous rate of change will be connected to the more general average rate of change.
Recall that for a moving object with position function s, its average velocity on the time
interval t = a to t = a + h is given by the quotient
AV [a,a+h] =
s(a + h) − s(a)
h
.
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
1.3 The derivative of a function at a point
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• How is the average rate of change of a function on a given interval defined, and
what does this quantity measure?
• How is the instantaneous rate of change of a function at a particular point defined?
How is the instantaneous rate of change linked to average rate of change?
• What is the derivative of a function at a given point? What does this derivative
value measure? How do we interpret the derivative value graphically?
• How are limits used formally in the computation of derivatives?
Introduction
An idea that sits at the foundations of calculus is the instantaneous rate of change of a
function. This rate of change is always considered with respect to change in the input
variable, often at a particular fixed input value. This is a generalization of the notion
of instantaneous velocity and essentially allows us to consider the question “how do we
measure how fast a particular function is changing at a given point?” When the original
function represents the position of a moving object, this instantaneous rate of change is
precisely velocity, and might be measured in units such as feet per second. But in other
contexts, instantaneous rate of change could measure the number of cells added to a
bacteria culture per day, the number of additional gallons of gasoline consumed by going
one mile per additional mile per hour in a car’s velocity, or the number of dollars added
to a mortgage payment for each percentage increase in interest rate. Regardless of the
presence of a physical or practical interpretation of a function, the instantaneous rate of
change may also be interpreted geometrically in connection to the function’s graph, and
this connection is also foundational to many of the main ideas in calculus.
In what follows, we will introduce terminology and notation that makes it easier
to talk about the instantaneous rate of change of a function at a point. In addition,
just as instantaneous velocity is defined in terms of average velocity, the more general
instantaneous rate of change will be connected to the more general average rate of change.
Recall that for a moving object with position function s, its average velocity on the time
interval t = a to t = a + h is given by the quotient
AV [a,a+h] =
s(a + h) − s(a)
h
.