Today weâ€™re going to take a look at one of most fundamental rules of calculus, the product rule. The product rule is used in many places so it is important that you know well. You will not be able to do more complicated calculus without having this essential tool. We're going to go through some in-depth examples so that the theory and application of this important rule are apparent. Keep in mind this is just a single example. You will have to find more if you are really struggling with this topic.

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**Example**

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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 February 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 pulling its existing subscribers the company found that each intended to install an average of 0.01 new phone lines by the end of February estimate the number of new lines of the company will have to install in February by counting the rate of increase of lines at the beginning of the month. This question seems complex, but it is quite easy if you break it down into components.

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Let s(t) be the number of subscribers and let n(t) be the number of phone lines per subscriber at time t. T is a unit of time that is one month. T=0 means the beginning of February. The total number of lines is;

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L(t) = s(t) * n(t) , we want to find Lâ€™(0). This can be done using the product rule

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Lâ€™(t) = sâ€™(t) * n(t) + nâ€™(t) * s(t)

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The question gives us the following information; s(0)= 100,000 and n(0)=1.2 , and sâ€™(0)=1000, nâ€™(0) = 0.01.

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So then:

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Lâ€™(0) = 1000 * 1.2 + 0.01 * 100,000 = 2200

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We can say that the company must install approximately 2200 new phone lines in February. You can see that the two different functions in this equation come from old users and new users.Â Hopefully you were not lost in this example problem, because it was fairly basic.This is a great example that shows the practical limits of the product rule. You will be seeing a lot more of this calculus rule, so make sure you are completely comfortable with it. Don't forget you can always ask your teacher for extra help.

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