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1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
The Derivative of a Function at a Point
Just as we defined instantaneous velocity in terms of average velocity, we now define
the instantaneous rate of change of a function at a point in terms of the average rate of
change of the function f over related intervals. In addition, we give a special name to “the
instantaneous rate of change of f at a,” calling this quantity “the derivative of f at a,”
with this value being represented by the shorthand notation f ′ (a). Specifically, we make
the following definition.
Definition 1.3. Let f be a function and x = a a value in the function’s domain. We define
the derivative of f with respect to x evaluated at x = a, denoted f ′ (a), by the formula
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
,
provided this limit exists.
Aloud, we read the symbol f ′ (a) as either “ f -prime at a” or “the derivative of f
evaluated at x = a.” Much of the next several chapters will be devoted to understanding,
computing, applying, and interpreting derivatives. For now, we make the following
important notes.
• The derivative of f at the value x = a is defined as the limit of the average rate of
change of f on the interval [a, a + h] as h → 0. It is possible for this limit not to
exist, so not every function has a derivative at every point.
• We say that a function that has a derivative at x = a is differentiable at x = a.
• The derivative is a generalization of the instantaneous velocity of a position function:
when y = s(t) is a position function of a moving body, s ′ (a) tells us the instantaneous
velocity of the body at time t = a.
• Because the units on
f (a+h)− f (a)
h
are “units of f per unit of x,” the derivative has
these very same units. For instance, if s measures position in feet and t measures
time in seconds, the units on s ′ (a) are feet per second.
• Because the quantity
f (a+h)− f (a)
h
represents the slope of the line through (a, f (a))
and (a + h, f (a + h)), when we compute the derivative we are taking the limit of a
collection of slopes of lines, and thus the derivative itself represents the slope of a
particularly important line.
While all of the above ideas are important and we will add depth and perspective to
them through additional time and study, for now it is most essential to recognize how the
derivative of a function at a given value represents the slope of a certain line. Thus, we
expand upon the last bullet item above.
1.3. THE DERIVATIVE OF A FUNCTION AT A POINT
The Derivative of a Function at a Point
Just as we defined instantaneous velocity in terms of average velocity, we now define
the instantaneous rate of change of a function at a point in terms of the average rate of
change of the function f over related intervals. In addition, we give a special name to “the
instantaneous rate of change of f at a,” calling this quantity “the derivative of f at a,”
with this value being represented by the shorthand notation f ′ (a). Specifically, we make
the following definition.
Definition 1.3. Let f be a function and x = a a value in the function’s domain. We define
the derivative of f with respect to x evaluated at x = a, denoted f ′ (a), by the formula
f
′ (a) = lim
h→0
f (a + h) − f (a)
h
,
provided this limit exists.
Aloud, we read the symbol f ′ (a) as either “ f -prime at a” or “the derivative of f
evaluated at x = a.” Much of the next several chapters will be devoted to understanding,
computing, applying, and interpreting derivatives. For now, we make the following
important notes.
• The derivative of f at the value x = a is defined as the limit of the average rate of
change of f on the interval [a, a + h] as h → 0. It is possible for this limit not to
exist, so not every function has a derivative at every point.
• We say that a function that has a derivative at x = a is differentiable at x = a.
• The derivative is a generalization of the instantaneous velocity of a position function:
when y = s(t) is a position function of a moving body, s ′ (a) tells us the instantaneous
velocity of the body at time t = a.
• Because the units on
f (a+h)− f (a)
h
are “units of f per unit of x,” the derivative has
these very same units. For instance, if s measures position in feet and t measures
time in seconds, the units on s ′ (a) are feet per second.
• Because the quantity
f (a+h)− f (a)
h
represents the slope of the line through (a, f (a))
and (a + h, f (a + h)), when we compute the derivative we are taking the limit of a
collection of slopes of lines, and thus the derivative itself represents the slope of a
particularly important line.
While all of the above ideas are important and we will add depth and perspective to
them through additional time and study, for now it is most essential to recognize how the
derivative of a function at a given value represents the slope of a certain line. Thus, we
expand upon the last bullet item above.