Math 124
Abegg
Handout (3.4)
Chain Rule Application Problems
Note: This homework assignment must be turned in on a separate sheet of paper.
Final answers are given to help you know if you are doing the problems correctly. Full solutions will be posted on Canvas.
If the quantity 𝐴 is given by a composition 𝐴(𝑔 𝑥 ), then the chain rule shows that the rate of change of 𝐴 with
respect to 𝑥 is calculated by multiplying the rate of change of 𝐴 with respect to 𝑔 with the rate of change of 𝑔
with respect to 𝑥. You will need to be comfortable with the following notations of the chain rule.
General derivative:
Derivative evaluated at a specific point:
𝑑𝐴 𝑑𝐴 𝑑𝑔
=
⋅
= 𝐴! 𝑔(𝑥 ) ⋅ 𝑔! 𝑥
𝑑𝑥 𝑑𝑔 𝑑𝑥
𝑑𝐴
𝑑𝑥
!!!
=
𝑑𝐴
𝑑𝑔
⋅
!!!(!)
𝑑𝑔
𝑑𝑥
!!!
= 𝐴! 𝑔 𝑎
⋅ 𝑔! 𝑎
1. Bacterial growth produces heat. The biomass (in grams) of a particular sample of bacteria
grows according to the equation 𝑚 𝑡 = 0.4 + 0.1𝑡 !.! , where 𝑡 is in minutes. The
temperature (in °C) of 𝑚 grams of biomass is given by 𝑇 𝑚 = 20 + 2𝑚. For each of the
following, find the value and interpret what it represents with regard to the situation.
𝑑𝑚
𝑑𝑇
a. 𝑚 15
b. 𝑚! (15) =
c. 𝑇(𝑚 15 )
d.
𝑑𝑡 !!!"
𝑑𝑡 !!!"
Ans: a. 3.78 grams; b. 0.29 grams/min; c. 22.7°C; d. 0.1°C/min
2. The population of a city is given by 𝑝 𝑡 = 8 + 𝑡𝑒 !!.!"! thousand people, where 𝑡 is the
number of years since 2015. The number of garbage trucks needed by the city can be modeled
by the equation 𝑔 𝑝 = 4 + 𝑝 + 5𝑝! , where 𝑝 is the population in thousands. Find the
value of each of the following in 2025. Include units in your answers.
𝑑𝑔
𝑑𝑝
𝑑𝑔
a. 𝑝(𝑡)
b. 𝑔(𝑝)
c.
d.
e.
𝑑𝑝
𝑑𝑡
𝑑𝑡
Ans: a. 14,060 people; b. 32 garbage trucks; c. 303 people/year; d. 2.2 trucks/thousand people; e. 0.68 trucks/year
3. A beehive with 𝑏 bees produces ℎ 𝑏 = 20(1 − 0.95! ) grams of honey each day. The price
!""
!"
of a gram of honey, in dollars, is given by 𝑃 ℎ = !!!. Find and interpret !"
!!!"
.
Ans: –1.4 cents per additional bee
# Rabbits
4. The graph shows the number of rabbits, 𝑅, living in a certain garden
𝑡 weeks from now (in reality the graph should be discrete points since you cannot
have fractional rabbits, but let's just consider it a continuous function for this problem).
The number of carrots in the garden, 𝐶, depends on the number of
rabbits: 𝐶 𝑅 = 300
!"!!
!"!!!
. Calculate the rate at which the
number of carrots in the garden is changing with time when:
a. 𝑡 = 1 week
b. 𝑡 = 5 weeks
c. 𝑅 = 5 rabbits
Handout (3.4) – Page 1 of 1
𝑡 (weeks)
Ans: a. –6.7; b. 0; c. 30 (units: carrots/week)