5128_Ch03_pp098-184

100 Chapter 3 Derivatives
f(a + h)
y
y = f(x)
f(a) P(a, f(a)) Q(a + h, f(a + h))
O
a
x
y
a + h
Figure 3.1 The slope of the secant line
PQ is
y
f a h
f a
x a h
a
f(x)
f(a)
O
y
P(a, f(a))
a
f a h f a
.
h
y = f(x)
Q(x, f(x))
x
y
Figure 3.2 The slope of the secant line
PQ is
y
f x f a
.
x x a
x
x
x
SOLUTION
Applying the definition, we have
f x lim f x h f x
h→0 h
lim x h 3 x
3
h→0 h
x 3 3x 2 h 3xh 2 h 3 x
lim
3
h→0
h
lim 3x2 3xh h 2 h
h→0 h
x h 3
expanded
x 3 s cancelled,
h factored out
lim
h→0
3x 2 3xh h 2 3x 2 . Now try Exercise 1.
The derivative of f x at a point where x a is found by taking the limit as h→0 of
slopes of secant lines, as shown in Figure 3.1.
By relabeling the picture as in Figure 3.2, we arrive at a useful alternate formula for
calculating the derivative. This time, the limit is taken as x approaches a.
DEFINITION (ALTERNATE) Derivative at a Point
The derivative of the function f at the point x a is the limit
f a lim f x f a
, (2)
x→a x a
provided the limit exists.
After we find the derivative of f at a point x a using the alternate form, we can find
the derivative of f as a function by applying the resulting formula to an arbitrary x in the
domain of f.
EXAMPLE 2
Differentiate f x x
SOLUTION
Applying the Alternate Definition
At the point x a,
f a lim f x f a
x→a x a
using the alternate definition.
lim x a
x→a x a
1
lim
x→a x a
Eq. 1 with fx x 3 ,
fx h x h 3
Eq. 2 with f x 1x
lim x
x→a x
a
a
• x a
Rationalize…
x a
x a
lim
…the numerator.
x→a x ax a
We can now take the limit.
1
.
2 a
Applying this formula to an arbitrary x 0 in the domain of f identifies the derivative as
the function f x 12x with domain 0, . Now try Exercise 5.