CHAPTER 5 DERIVATIVES
LEARNING
OBJECTIVES
Upon
completion of this chapter, you should be able to do the following:
1.
Compute the derivative of a constant.
2.
Compute the derivative of a variable raised to a power.
3.
Compute the derivative of the sum and product of two or more functions and the
quotient of two functions.
4.
Compute the derivative of a function raised to a power, in radical form, and by
using the chain rule.
5.
Compute the derivative of an inverse function, an implicit function, a
trigonometric function, and a natural logarithmic function.
6.
Compute the derivative of a constant raised to a variable power.
INTRODUCTION
In
the previous chapter on limits, we used the delta process to find the limit of
a function as Ax approached zero. We called the result of this tedious and, in
some cases, lengthy process the derivative. In this chapter we will examine
some rules used to find the derivative of a function without using the delta
process. To find how y changes as x changes, we take the limit of
which is called the derivative of y with respect to x; we
use the symbol
to indicate the derivative and
write
In this section we will learn a number of rules that will
enable us to easily obtain the derivative of many algebraic functions. In the
derivation of these rules, which will be called theorems, we will assume that
or
exists and is finite.
DERIVATIVE OF A CONSTANT
The method we will use to find the derivative of a
constant is similar to the delta process used in the previous chapter but includes
an analytical proof. A diagram is used to give a geometrical meaning of the
function.
Theorem 1. The derivative of a constant is zero. Expressed as a formula, this may be written as
where y = c.
PROOF: In figure 51, the graph of
Figure 51.Graph of y = c, where c is a constant.
y=c
where c is a constant, the value of y is the same for all
values of x, and any change in x (that is,
x) does not
affect y; then
and
Another way of stating this is that when x is equal to x_{1}
and when x is equal to
has the same value. Therefore,
y=c
and
so that
and
Then
The equation
represents a straight line parallel to the X axis. The
slope of this line will be zero for all values of x. Therefore, the derivative
is zero for all values of x.
EXAMPLE.
Find the derivative
of the
function
SOLUTION:
and
Therefore,
