1. ## 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.

2. Originally Posted by hemi
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$\displaystyle =x'=\frac{dx}{dp}=\frac{d}{dp}\left[\frac{500}{2p^2+1}\right]=\frac{(2p^2+1)0-500(4p)}{(2p^2+1)^2}$