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Thread: rate of change

  1. #1
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    rate of change

    Find the rate of change of change of x with respect to p:

    $\displaystyle p=\sqrt \frac {500-x}{2x}$ , $\displaystyle 0<x<500$

    If you could guide me through this problem that would be great.
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by hemi View Post
    Find the rate of change of change of x with respect to p:

    $\displaystyle p=\sqrt \frac {500-x}{2x}$ , $\displaystyle 0<x<500$

    If you could guide me through this problem that would be great.

    rate of change can be viewed as slope geometrically. now certainly you want the instantaneous rate of change of x with respect to p.

    ie $\displaystyle \frac{dx}{dp}$ known as the derivative of x with respect to p.

    so let's find the slope. remember that the p is shown respect here so it is our variable.

    im not sure how far into calc you are but i'll do it the long way

    $\displaystyle p^2=\frac{500-x}{2x}$

    $\displaystyle 2p^2x+x=500$

    $\displaystyle x(2p^2+1)=500$

    $\displaystyle x=\frac{500}{2p^2+1}$

    now we take the derivative

    inst. rate of change$\displaystyle =x'=\frac{dx}{dp}=\frac{d}{dp}\left[\frac{500}{2p^2+1}\right]=\frac{(2p^2+1)0-500(4p)}{(2p^2+1)^2}$
    Last edited by mr fantastic; Jun 12th 2009 at 04:07 AM. Reason: Made the square brackets bigger
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