18.6 Calculus 617

During the next two years, as you take your calculus classes, you will learn many new con-

cepts and rules dealing with differential calculus. Make sure you take the time to understand

these concepts and rules. In your calculus classes, you may not apply the concepts to actual

engineering problems, but be assured that you will use them eventually in your engineering

classes. Some of these concepts and rules are summarized in Table 18.6. Examples that dem-

onstrate how to apply these differentiation rules follow. As you study these examples, keep in

mind that our intent here is to introduce some rules, not to explain them thoroughly.

TABLE 18.6 Summary of Definitions and Derivative Rules

Definitions and Rules Explanation

1 The definition of the derivative of the

functionf(x).

2Iff(x) constant thenf¿(x)  0 The derivative of a constant function is zero.

3Iff(x) x

n

thenf¿(x) nx

n 1

The Power Rule (see Example 18.5).

4Iff(x) ag(x) whereais a constant The rule for when a constant such asais

thenf¿(x) ag¿(x) multiplied by a function (see Example 18.6).

5Iff(x) g(x) h(x) thenf¿(x)  The rule for when two functions are added

g¿(x) h¿(x) or subtracted (see Example 18.7).

6Iff(x) g(x) h(x) thenf¿(x)  The Product Rule (see Example 18.8).

g¿(x) h(x) g(x)h¿(x)

7 If then The Quotient Rule (see Example 18.9).

8Iff(x) f[g(x)] f(u) whereug(x) The Chain Rule.

then

9Iff(x) [g(x)]

n

u

n

whereug(x) The Power Rule for a general function such

then

asg(x) (see Example 18.10).

10 Iff(x) ln |g(x)| The rule for natural logarithm functions

then

(see Example 18.11).

11 Iff(x) exp(g(x)) orf(x) e

g(x)

The rule for exponential functions

thenf¿(x) g¿(x) e

g(x)

(see Example 18.12).

f

œ

1 x 2 

g

œ

1 x 2

g 1 x 2

f

œ

1 x 2 n#u

n  (^1) #du
dx
f
œ
1 x 2 
df 1 x 2
dx

df 1 x 2
du

du
dx
f
œ
1 x 2 
h 1 x 2 #g
œ
1 x 2 g 1 x 2 #h
œ
1 x 2
3 h 1 x 24
2
f 1 x 2 
g 1 x 2
h 1 x 2
f
œ
1 x 2 
df
dx
limit
h S 0
f 1 xh 2 f 1 x 2
h
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