2.5. THE CHAIN RULE
125
Activity 2.14.
For each of the following functions, find the function’s derivative. State the rule(s) you
use, label relevant derivatives appropriately, and be sure to clearly identify your overall
answer.
(a) p(r) = 4
√
r 6 + 2e r
(b) m(v) = sin(v 2 ) cos(v 3 )
(c) h(y) =
cos(10y)
e 4y + 1
(d) s(z) = 2 z 2 sec(z)
(e) c(x) = sin(e
x 2 )
⊳
The chain rule now adds substantially to our ability to do different familiar problems
that involve derivatives. Whether finding the equation of the tangent line to a curve,
the instantaneous velocity of a moving particle, or the instantaneous rate of change of
a certain quantity, if the function under consideration involves a composition of other
functions, the chain rule is indispensable.
Activity 2.15.
Use known derivative rules, including the chain rule, as needed to answer each of the
following questions.
(a) Find an equation for the tangent line to the curve y =
√
e x + 3 at the point
where x = 0.
(b) If s(t) =
1
(t 2 + 1) 3 represents the position function of a particle moving horizontally along an axis at time t (where s is measured in inches and t in seconds),
find the particle’s instantaneous velocity at t = 1. Is the particle moving to the
left or right at that instant?
(c) At sea level, air pressure is 30 inches of mercury. At an altitude of h feet above
sea level, the air pressure, P, in inches of mercury, is given by the function
P = 30e
−0.0000323h .
Compute dP/dh and explain what this derivative function tells you about air
pressure, including a discussion of the units on dP/dh. In addition, determine
how fast the air pressure is changing for a pilot of a small plane passing through
an altitude of 1000 feet.
(d) Suppose that f (x) and g(x) are differentiable functions and that the following
information about them is known:
125
Activity 2.14.
For each of the following functions, find the function’s derivative. State the rule(s) you
use, label relevant derivatives appropriately, and be sure to clearly identify your overall
answer.
(a) p(r) = 4
√
r 6 + 2e r
(b) m(v) = sin(v 2 ) cos(v 3 )
(c) h(y) =
cos(10y)
e 4y + 1
(d) s(z) = 2 z 2 sec(z)
(e) c(x) = sin(e
x 2 )
⊳
The chain rule now adds substantially to our ability to do different familiar problems
that involve derivatives. Whether finding the equation of the tangent line to a curve,
the instantaneous velocity of a moving particle, or the instantaneous rate of change of
a certain quantity, if the function under consideration involves a composition of other
functions, the chain rule is indispensable.
Activity 2.15.
Use known derivative rules, including the chain rule, as needed to answer each of the
following questions.
(a) Find an equation for the tangent line to the curve y =
√
e x + 3 at the point
where x = 0.
(b) If s(t) =
1
(t 2 + 1) 3 represents the position function of a particle moving horizontally along an axis at time t (where s is measured in inches and t in seconds),
find the particle’s instantaneous velocity at t = 1. Is the particle moving to the
left or right at that instant?
(c) At sea level, air pressure is 30 inches of mercury. At an altitude of h feet above
sea level, the air pressure, P, in inches of mercury, is given by the function
P = 30e
−0.0000323h .
Compute dP/dh and explain what this derivative function tells you about air
pressure, including a discussion of the units on dP/dh. In addition, determine
how fast the air pressure is changing for a pilot of a small plane passing through
an altitude of 1000 feet.
(d) Suppose that f (x) and g(x) are differentiable functions and that the following
information about them is known: