In this video I go over a useful word problem on using the product rule for derivatives in estimating the amount of phone lines a phone company needs to install in the next month.

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Product Rule for Derivatives: Example Word Problem

Product Rule Recap

If f and g are both differentiable, then:

Example

A telephone company wants to estimate the number of new residential phone lines that it will need to install during the upcoming month.

At the beginning of January the company had 100,000 subscribers, each of whom had 1.2 phone lines, on average.

The company estimated that its subscribership was increasing at the rate of 1000 monthly. By polling its existing subscribers, the company found that each intended to install an average of 0.01 new phone lines by the end of January.

Estimate the number of new lines the company will have to install in January by calculating the rate of increase of lines at the beginning of the month.

Solution

Let s(t) be the number of subscribers.
Let n(t) be the number of phone line per subscriber.
Let L(t) be the total number of subscribers.

t is measured in months and t = 0 is at the beginning of January.