- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
The chain rule and logarithms practice question 18
The revenue from selling units of a product is dollars. Sales are growing: weeks from now the company expects to be selling units. How fast is the revenue changing 5 weeks from now?
Give the answer in dollars per week.
Revenue depends on , and depends on . How do you combine a rate in dollars per unit with a rate in units per week?
The chain rule gives . Find each factor separately.
Evaluate at the sales level 5 weeks from now, , not at 5.
An exact value.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Revenue is a function of sales, and sales are a function of time, so revenue is a composition of the two. The chain rule multiplies the two rates, with dR/dq read at the sales level of the moment.
Working
Differentiate each link of the chain.
The sales level at the moment asked about.
Chain rule, with dR/dq read at q = 40.
Answer
The answer is the revenue is rising at 80 dollars per week.
Check
Between weeks 4.5 and 5.5, sales go from 38 to 42 units and revenue from 1558 to 1638 dollars, a rise of 80 dollars over that week.
Where this goes wrong
Putting t = 5 into dR/dq, which gives (60 − 5)(4) = 220. dR/dq = 60 − q is a function of q, so it is evaluated at the sales level q = 40.
Leans on
The power rule, from Differentiation rules
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