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Derivatives
The Derivative at a
Point
The derivative of a function
at a point is its instantaneous rate of change at that point. It
is also the slope of the tangent line to the graph of the function
at that point (provided that the tangent line exists and is not
vertical). Given a function, _{ }
we can use the notation
_{}, to denote the derivative of f with
respect to t at the point t = a or the instantaneous
rate of change of f at t = a.
Example
1
Consider the function
C(t )= .0217t ² 0.9205t + 13.0983, which can be used to model
the population (in thousands) of Chattahoochee County, Ga in the
year 2000 + t. Interpret the following statements in the context
of this situation.
(a)
C′(25)= 0.165 thousand people per year
(b)
C′(0)= 0.921 thousand people per year
(c)
The derivative of C with respect to t is 0.487 thousand people
per year when
t = 10.
Solution
(a)
This statement says that the population of Chattahoochee County
was increasing by 165 people per year at the beginning of 1975.
(b)
This means that in the beginning of the year 2000, the population
of Chattahoochee County was decreasing by 921 people per year.
(c)
The third statement tells us that the population of Chattahoochee
County was decreasing by 487 people per year in the beginning of
1990.
Notice that all of these statements tell us something about how
the population of Chattahoochee County was changing at a particular
time. They do not, however, tell us what the population was at that
time. We can get that information from the function C(t).
Example 2
Figure
1 below shows the graph of a function f(t) and its tangent line
at the point
t = 1. Estimate the
values of f(1) and f′(1).
Figure 1
Solution
Since
f(1) is the ycoordinate of the point on the graph of f where
t = 1, we see that f(1) = 2. To find f′(1), we must estimate
the slope of the tangent line at the point t = 1. We choose two
points on the tangent line, (2.5,0) and (1,2), and calculate its
slope:
_{}
So,
f′(1) ≈ 1.33.
Example 3
If
f(x) = x²2x+1, estimate the value of f′(2) .
Solution
Since
f′(2) is the slope of the tangent line to the graph of f(x)
at the point x = 2, we can use the graphical method to find f′(2).That
is we graph the function and zoomin on the point x = 2 until the
graph looks like its tangent line. Figure 2 shows a closeup view
of the part of the parabola containing the point x = 2.
Figure 2
We see
that a good estimate of the slope at the point x = 2 can be obtained
by choosing the two points (1.9, 0.8) and (2,1). Our estimate is
_{}
This means that the slope
of the tangent line to the graph of f(x) = x²2x+1 at the point
(2,1) is approximately 2.0.
Note: You could choose
any two points on the above graph and get an answer that is close
to 2.0.
The Derivative Function
Since the derivative of a function at a point is the slope or rate
of change of the function at that point, the derivative takes on
exactly one value for each different value of x in the domain of
f (as long as the tangent line exists and is nonvertical at that
point). Therefore, the derivative is also a function. Its domain
is the set of all numbers in the domain of f for which the graph
of f has a nonvertical tangent line. There are several different
notations for the derivative of a function. Three of the most commonly
used notations are given below.
If y = f(x) is a function, then
(1) f′(x), read “f prime of x”, denotes the derivative
of f with respect to x;
(2)
_{} denotes the derivative of y with respect to
x;
(3) _{} denotes the derivative of f with respect to
x.
We can use the following
derivativedefinition to find f′(x) for any given function
f(x):
Derivative
Definition
_{}
Example 1
Find the derivative of
the function _{}
Solution
_{}
Substituting into the
derivative formula gives,
Basic Derivative Rules
The following differentiation
formulas can be used to find the derivative of many basic algebraic
functions.

Derivative of a
Linear Function
If _{}
Example
Find the derivative of
the linear function _{}
Solution
_{}

The Constant Rule
_{}
Example
Find the derivative of
the constant function _{}
Solution
_{}

The Power Rule
_{}
Example
Find the derivative of
each of the functions _{}
Solution
(i)_{}; (ii) _{}; (iii) _{}

The Constant Multiplier
Rule
If _{}
Example
Find the derivative of
the function _{}
Solution
Using the Constant Multiplier
Rule and the Power Rule, we get
_{}

The Sum Rule
_{}
Example
Find the derivative of
the function _{}
Solution
Applying the Sum, Constant
Multiplier, Power, and Constant Function Rules gives
_{}

The Product Rule
If _{}
Example
Find the derivative of
the function _{}
Solution
_{}

The Quotient Rule
If _{}
Example
Find the derivative of
the function _{}
Solution
_{}

The Chain Rule
_{}
Example
Find the derivative of
the function _{}
Solution
Using the chain rule
gives
_{}