11DIFFERENTIATION - Department of Mathematics

694 11 DIFFERENTIATION
11.1 Basic Rules of Differentiation
Rule 1: Derivative
of a Constant
FIGURE 11.1
The slope of the tangent line to the graph
of f(x) c, where c is a constant, is zero.
y
f(x) = c
x
F OUR B ASIC R ULES
The method used in Chapter 10 for computing the derivative of a function is
based on a faithful interpretation of the definition of the derivative as the
limit of a quotient. Thus, to find the rule for the derivative f of a function f,
we first computed the difference quotient
f(x h) f(x)
h
and then evaluated its limit as h approached zero. As you have probably
observed, this method is tedious even for relatively simple functions.
The main purpose of this chapter is to derive certain rules that will simplify
the process of finding the derivative of a function. Throughout this book we
will use the notation
d
dx [f(x)]
[read ‘‘d, dxoffofx’’] to mean ‘‘the derivative of f with respect to x at x.’’
In stating the rules of differentiation, we assume that the functions f and g
are differentiable.
d
(c) 0 (c, a constant)
dx
The derivative of a constant function is equal to zero.
We can see this from a geometric viewpoint by recalling that the graph
of a constant function is a straight line parallel to the x-axis (Figure 11.1).
Since the tangent line to a straight line at any point on the line coincides with
the straight line itself, its slope [as given by the derivative of f(x) c] must
be zero. We can also use the definition of the derivative to prove this result
by computing
f(x) lim
h0
c c
lim
h0 h
lim 0 0
h0
f(x h) f(x)
h