5128_Ch03_pp098-184
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100 Chapter 3 Derivatives<br />
f(a + h)<br />
y<br />
y = f(x)<br />
f(a) P(a, f(a)) Q(a + h, f(a + h))<br />
O<br />
a<br />
x<br />
y<br />
a + h<br />
Figure 3.1 The slope of the secant line<br />
PQ is<br />
y<br />
f a h<br />
f a<br />
<br />
x a h<br />
a<br />
f(x)<br />
f(a)<br />
O<br />
y<br />
P(a, f(a))<br />
a<br />
f a h f a<br />
.<br />
h<br />
y = f(x)<br />
Q(x, f(x))<br />
x<br />
y<br />
Figure 3.2 The slope of the secant line<br />
PQ is<br />
y<br />
f x f a<br />
.<br />
x x a<br />
x<br />
x<br />
x<br />
SOLUTION<br />
Applying the definition, we have<br />
f x lim f x h f x<br />
<br />
h→0 h<br />
lim x h 3 x<br />
<br />
3<br />
h→0 h<br />
x 3 3x 2 h 3xh 2 h 3 x<br />
lim <br />
3<br />
h→0<br />
h<br />
lim 3x2 3xh h 2 h<br />
<br />
h→0 h<br />
x h 3<br />
expanded<br />
x 3 s cancelled,<br />
h factored out<br />
lim<br />
h→0<br />
3x 2 3xh h 2 3x 2 . Now try Exercise 1.<br />
The derivative of f x at a point where x a is found by taking the limit as h→0 of<br />
slopes of secant lines, as shown in Figure 3.1.<br />
By relabeling the picture as in Figure 3.2, we arrive at a useful alternate formula for<br />
calculating the derivative. This time, the limit is taken as x approaches a.<br />
DEFINITION (ALTERNATE) Derivative at a Point<br />
The derivative of the function f at the point x a is the limit<br />
f a lim f x f a<br />
, (2)<br />
x→a x a<br />
provided the limit exists.<br />
After we find the derivative of f at a point x a using the alternate form, we can find<br />
the derivative of f as a function by applying the resulting formula to an arbitrary x in the<br />
domain of f.<br />
EXAMPLE 2<br />
Differentiate f x x<br />
SOLUTION<br />
Applying the Alternate Definition<br />
At the point x a,<br />
f a lim f x f a<br />
<br />
x→a x a<br />
using the alternate definition.<br />
lim x a<br />
<br />
x→a x a<br />
1<br />
lim<br />
x→a x a<br />
Eq. 1 with fx x 3 ,<br />
fx h x h 3<br />
Eq. 2 with f x 1x<br />
lim x <br />
x→a x <br />
a<br />
a<br />
• x a<br />
<br />
Rationalize…<br />
x a<br />
x a<br />
lim<br />
…the numerator.<br />
x→a x ax a<br />
We can now take the limit.<br />
1<br />
.<br />
2 a<br />
Applying this formula to an arbitrary x 0 in the domain of f identifies the derivative as<br />
the function f x 12x with domain 0, . Now try Exercise 5.