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Thread: Derivative question

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    Derivative question

    This may sound a bit confusing.
    If there's a problem, that says: "A manufacturer finds that the daily profit, $P, from selling n articles is given by P= 100n - 0.4n^2 - 160 ", would the 'profit per article' be the derivative dP/dn, or would it be P/n? And if you want to find the value of n which maximises the profit per article, would you use the double derivative, or would you just find the maximum of the derivative of P/n?
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    Re: Derivative question

    $\frac{dP}{dn}$ is the marginal profit per article, it is the additional profit you would get from producing one more article (or rather a continuous approximation to it). The profit per article is ${P}/{n}$

    You find the $n$ that maximises the profit per article by finding the stationary points of $P/N$ where $\frac{d}{dn}(P/n)=0$ and choosing the one coresponding to maximum profit per article if there is more than one solution.

    In this case there are two solutions to $\frac{d}{dn}(P/n)=0$ but one of them is negative and so not possible.
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    Re: Derivative question

    So dP/dn is the additional profit one would gain from producing one more article, then if you let n=2, would dP/dn give the additional profit from producing a second article from the first, or a third article from the second? And if you calculate the profit using P/n of producing 3 articles, minus the profit of producing 2 articles, would you get the same answer?
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    Re: Derivative question

    Quote Originally Posted by winkyinky146 View Post
    So dP/dn is the additional profit one would gain from producing one more article, then if you let n=2, would dP/dn give the additional profit from producing a second article from the first, or a third article from the second? And if you calculate the profit using P/n of producing 3 articles, minus the profit of producing 2 articles, would you get the same answer
    You can read the page Austrian school from which the term marginal is derived. In analysis $\dfrac{dP}{dn}$ is a derivative; it is a measure of the rate of change in the profit with respect to number sold.
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    Re: Derivative question

    Quote Originally Posted by Plato View Post
    You can read the page Austrian school from which the term marginal is derived. In analysis $\dfrac{dP}{dn}$ is a derivative; it is a measure of the rate of change in the profit with respect to number sold.
    I see, so the derivative kind of tells you how dramatically the profit changes rather than the change in profit?
    I think I sort of understand..., If there is a problem that goes like, a car travels in a way such that distance from a point (D) = t^3 - 6t^2 +9t , where t is time. Wouldn't the derivative dD/dt, which is velocity, be basically the same as d/t? According to what you said, the derivative dD/dt would be a measure of the change in velocity over time, then if I were to find the maximum velocity of the car would I calculate the maximum of the derivative of D/t rather than dD/dt? And if I were to find the times where the velocity is zero, would I let 0=dD/dt or let 0 = D/t?
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