618 Chapter 18 Mathematics in Engineering
Example 18.4 Identify the dependent and independent variables for the following situations: water con-
sumption and traffic flow.
For the water consumption situation, the mass or the volume is the dependent variable,
with time the independent variable. For the traffic flow problem, the number of cars is the
dependent variable and time is the independent variable.Example 18.5 Find the derivative off(x) x
3
10 x
2
8.
We use rules 3 and 5,f¿(x) nx
n 1
, from Table 18.6 to solve this problem as shown.Example 18.6 Find the derivative off(x) 5(x
3
10 x
2
8).
We use rule 4,f¿(x) a#g¿(x), from Table 18.6 to solve this problem. For the given
problem, ifa 5 andg(x) x
3
10 x
2
8, then the derivative off(x) isf¿(x)
5(3x
2
20 x) 15 x
2
100 x.Example 18.7 Find the derivative off(x) (x
3
10 x
2
8) (x
5
5 x).
We use rule 5,f¿(x) g¿(x) h¿(x), from Table 18.6 to solve this problem as shown.Example 18.8 Find the derivative off(x) (x
3
10 x
2
8)(x
5
5 x).
We use rule 6,f¿(x) g¿(x) #h(x) g(x) #h¿(x), from Table 18.6 to solve this problem as
shown. For this problem,h(x) (x
5
5 x) andg(x) (x
3
10 x
2
8).Example 18.9 Find the derivative of f(x)(x
3
10 x
2
8) /(x
5
5 x).
We use rule 7, f¿(x)[h(x)#g¿(x)g(x)#h¿(x)]/[h(x)]
2
from Table 18.6 to solve this
problem as shown. For this problem,h(x) (x
5
5 x) andg(x) (x
3
10 x
2
8).f
œ
1 x 2 1 x
5
5 x 213 x
2
20 x 2 1 x
3
10 x
2
8215 x
4
521 x
5
5 x 2
2 8 x
7
70 x
6
40 x
4
20 x
3
150 x
2
40f
œ
1 x 2 13 x
2
20 x 21 x
5
5 x 2 1 x
3
10 x
2
8215 x
4
52f
œ
1 x 2 13 x
2
20 x 2 15 x
4
52f
œ
1 x 2 3 x
2
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