5128_Ch03_pp098-184

Section 3.3 Rules for Differentiation 121
[–3, 3] by [–2, 2]
Figure 3.19 The graph of
4x
y x
2
1 2
and the graph of
y NDER ( x x
2
2
)
1
1
appear to be the same. (Example 5)
SOLUTION
We apply the Quotient Rule with u x 2 1 and v x 2 1:
f x
2x 3 2x 2x 3 2x
x 2 1 2
4x
x
2
.
1 2
The graphs of y 1 f x calculated above and of y 2 NDER f x are shown in
Figure 3.19. The fact that they appear to be identical provides strong graphical support
that our calculations are indeed correct. Now try Exercise 19.
EXAMPLE 6
Working with Numerical Values
Let y uv be the product of the functions u and v. Find y2 if
u2 3, u2 4, v2 1, and v2 2.
SOLUTION
x 2 1 • 2x x 2 1 • 2x
x 2 1 2
From the Product Rule, yuvuvvu. In particular,
v(du dx) u(dv/dx)
v
2
y2 u2v2 v2u2
32 14
2. Now try Exercise 23.
Negative Integer Powers of x
The rule for differentiating negative powers of x is the same as Rule 2 for differentiating
positive powers of x, although our proof of Rule 2 does not work for negative values of n.
We can now extend the Power Rule to negative integer powers by a clever use of the Quotient
Rule.
RULE 7
Power Rule for Negative Integer Powers of x
If n is a negative integer and x 0, then
d
x d x
n nx n1 .
Proof of Rule 7 If n is a negative integer, then n m, where m is a positive integer.
Hence, x n x m 1x m , and
x
d
x d x
n d
d
( x
1
x m) m d d
• 1 1 • x d x d x
m
x m 2
0 x
mx
m1
2m
mx m1
nx n1 .
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