11DIFFERENTIATION - Department of Mathematics

It is estimated that t mo from now, the average price of
a PC will be given by
p(t) 400
200 (0 t 60)
1 t
dollars. Find the rate at which the quantity demanded
per month of the PCs will be changing 16 mo from now.
77. C RUISE S HIP B OOKINGS The management of Cruise World,
operators of Caribbean luxury cruises, expects that the
percentage of young adults booking passage on their
cruises in the years ahead will rise dramatically. They
have constructed the following model, which gives the
percentage of young adult passengers in year t:
p f(t) 50t2 2t 4
t2 (0 t 5)
4t 8
Young adults normally pick shorter cruises and generally
spend less on their passage. The following model gives
an approximation of the average amount of money R
(in dollars) spent per passenger on a cruise when the
percentage of young adults is p:
p 4
R(p) 1000p 2
Find the rate at which the price of the average passage
will be changing 2 yr from now.
In Exercises 78–81, determine whether the
statement is true or false. If it is true, explain
why it is true. If it is false, give an example to
show why it is false.
78. If f and g are differentiable and h f g, then
h(x) f[g(x)]g(x).
S OLUTIONS TO S ELF-CHECK E XERCISES 11.3
1. Rewriting, we have
Using the general power rule, we find
11.3 THE CHAIN RULE 733
79. If f is differentiable and c is a constant, then
d
[f(cx)] cf(cx).
dx
80. If f is differentiable, then
d f(x)
f(x)
dx 2f(x)
81. If f is differentiable, then
d
dxf1 x f 1
x
82. In Section 11.1 we proved that
d
dx (xn ) nx n1
for the special case when n 2. Use the chain rule to
show that
d
dx (x1/n ) 1
n x1/n1
for any nonzero integer n, assuming that f(x) x 1/n is
differentiable.
Hint: Let f(x) x 1/n so that [ f(x)] n x. Differentiate both sides
with respect to x.
83. With the aid of Exercise 82, prove that
d
dx (xr ) rx r1
for any rational number r.
Hint: Let r m/n, where m and n are integers with n 0, and
write x r (x m ) 1/n .
f(x) (2x 2 1) 1/2
f(x) d
dx (2x2 1) 1/2
2 (2x 2 1)
1
3/2 d
dx (2x2 1)
1
2 (2x2 1) 3/2 (4x)
2x
(2x2 1) 3/2