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Math Help - Rate Problem

  1. #1
    Junior Member
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    Sep 2009
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    Question Rate Problem

    Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate the derivative when x = 5 mm.

    V '(5) = ?


    What does this quantity represent?

    V '(5) represents the rate at which the volume is increasing with respect to the side length as V reaches 15 mm3.
    V '(5) represents the rate at which the volume is increasing as x reaches 15 mm.
    V '(5) represents the rate at which the volume is increasing with respect to the side length as x reaches 5 mm.
    V '(5) represents the rate at which the side length is increasing with respect to the volume as x reaches 5 mm.
    V '(5) represents the volume as the side length reaches 5 mm.

    (Attempt)

    I dont' know wat to do? Do I need to do the derivatives? Please show me! Thank you!

    Last edited by miss.strw; November 9th 2009 at 10:25 AM.
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  2. #2
    Senior Member
    Joined
    Dec 2008
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    First, we use the formula for the volume of a cube:

    V=x^3.

    We are asked to find the derivative of V with respect to x and evaluate it at x=5. To do this, we may either use the Power Rule:

    \frac{d}{dx}x^n=nx^{n-1}

    or we may use the definition of derivative:

    \frac{d}{dx}f(x)=\lim_{\small h\rightarrow 0}\frac{f(x+h)-f(x)}{h}.

    The derivative measures the rate of change of the function with respect to its independent variable. If V is considered a function of x, then the derivative of V basically measures the relative change of V when we change x by a very small amount.
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