Can't find derivative.

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August 6th 2012, 04:19 AM
viper483

Can't find derivative.

Can somebody tell me how to find the rate of change to this?

p = 240(1 - [3 / {3 + e^(-0.0005x)}]

I need to find the rate of change when x = 1000.

Thanks
August 6th 2012, 04:40 AM
Prove It

Re: Can't find derivative.

Quote:

Originally Posted by viper483

Can somebody tell me how to find the rate of change to this?

p = 240(1 - [3 / {3 + e^(-0.0005x)}]

I need to find the rate of change when x = 1000.

Thanks

Just so we're clear, is this your function?

$\displaystyle \begin{align*} p = 240\left(1 - \frac{3}{3 + e^{-0.0005x}}\right) \end{align*}$
August 6th 2012, 04:41 AM
viper483

Re: Can't find derivative.

Quote:

Originally Posted by Prove It

Just so we're clear, is this your function?

$\displaystyle \begin{align*} p = 240\left(1 - \frac{3}{3 + e^{-0.0005x}}\right) \end{align*}$

Yes, that's correct. I need to learn how to set it out like that!
August 6th 2012, 04:53 AM
Prove It

Re: Can't find derivative.

Quote:

Originally Posted by viper483

Yes, that's correct. I need to learn how to set it out like that!

$\displaystyle \begin{align*} p &= 240\left(1 - \frac{3}{3 + e^{-0.0005x}}\right) \\ &= 240 - \frac{720}{3 + e^{-0.0005x}} \\ &= 240 - 720\left(3 + e^{-0.0005x}\right)^{-1} \end{align*}$

You should be able to apply the Chain Rule now.
August 6th 2012, 05:18 AM
viper483

Re: Can't find derivative.

Quote:

Originally Posted by Prove It

$\displaystyle \begin{align*} p &= 240\left(1 - \frac{3}{3 + e^{-0.0005x}}\right) \\ &= 240 - \frac{720}{3 + e^{-0.0005x}} \\ &= 240 - 720\left(3 + e^{-0.0005x}\right)^{-1} \end{align*}$

You should be able to apply the Chain Rule now.

Am I supposed to use the chain rule with the exponent of e as in f(p)=-0.0005x and use that as the inner function?

My teacher said that when using the chain rule with an exponential function like this we're supposed to use the equation d/dx(e^f(x)) = e^f(x) * f '(x).

I'm trying to use that for this equation but I don't think I'm supposed to because the answer doesn't make sense.
August 6th 2012, 05:34 AM
HallsofIvy

Re: Can't find derivative.

There is no way for anyone to know where your answer makes sense or not if you don't tell us what answer you got!
August 6th 2012, 05:58 AM
Prove It

Re: Can't find derivative.

Quote:

Originally Posted by viper483

Am I supposed to use the chain rule with the exponent of e as in f(p)=-0.0005x and use that as the inner function?

My teacher said that when using the chain rule with an exponential function like this we're supposed to use the equation d/dx(e^f(x)) = e^f(x) * f '(x).

I'm trying to use that for this equation but I don't think I'm supposed to because the answer doesn't make sense.

I mean you let $\displaystyle \begin{align*} u = 3 + e^{-0.0005x} \end{align*}$ which gives $\displaystyle \begin{align*} y = 240 - 720u^{-1} \end{align*}$

You should be able to evaluate $\displaystyle \begin{align*} \frac{du}{dx} \end{align*}$ and $\displaystyle \begin{align*} \frac{dy}{du} \end{align*}$ then multiply them together...
August 6th 2012, 03:26 PM
viper483

Re: Can't find derivative.

Quote:

Originally Posted by Prove It

I mean you let $\displaystyle \begin{align*} u = 3 + e^{-0.0005x} \end{align*}$ which gives $\displaystyle \begin{align*} y = 240 - 720u^{-1} \end{align*}$

You should be able to evaluate $\displaystyle \begin{align*} \frac{du}{dx} \end{align*}$ and $\displaystyle \begin{align*} \frac{dy}{du} \end{align*}$ then multiply them together...

Sorry, it was midnight here last night, so ended up going to bed.

Ok, so if it's $\displaystyle \begin{align*} \frac{du}{dx} \end{align*}$ then

$u(1000)=3 + e^{-0.0005{1000}}$ which is $3.60653066...$

Using $\displaystyle \begin{align*} \frac{dy}{du} \end{align*}$ it has to be

$y(u)=240 - 720 (3.60653066)^{-1}$
$= 240 - 720 (0.277274781)$
$=40.36$

Is this correct? The rate of change I should be getting is -0.0168?