11DIFFERENTIATION - Department of Mathematics

11.2 THE PRODUCT AND QUOTIENT RULES 711
SOLUTION ✔ Rewrite h(x) in the form h(x) x1/2
x 2 . By the quotient rule, we find
1
EXAMPLE 6
SOLUTION ✔
(x
h(x)
2 1) d
dx (x1/2 ) x
(x 2 1) 2
1/2 d
(x
2 1)1 2 x1/2 x1/2 (2x)
(x 2 1) 2
1
2
x1/2 (x 2 1 4x 2 )
(x 2 1) 2
2 1 3x
APPLICATIONS
dx (x 2 1)
(Factoring out x 1/2
from the numerator)
2x(x 2 1) 2
The sales (in millions of dollars) of a laser disc recording of a hit movie t
years from the date of release is given by
S(t) 5t
t 2 1
a. Find the rate at which the sales are changing at time t.
b. How fast are the sales changing at the time the laser discs are released
(t 0)? Two years from the date of release?
a. The rate at which the sales are changing at time t is given by S(t). Using
the quotient rule, we obtain
S(t) d
dt 5t
t2 5
1 d
dt t
t2 1
5 (t2 1)(1) t(2t)
(t 2 1) 2
5t2 1 2t 2
(t2 1) 2
5(1 t2 )
(t2 1) 2
b. The rate at which the sales are changing at the time the laser discs are
released is given by
S(0)
5(1 0)
5 2 (0 1)
That is, they are increasing at the rate of $5 million per year.