Derivative of 14x/(1+12.6e^(-0.73x)

$\displaystyle f(x)=\frac{14x}{(1+12.6e^{-0.73x})}$

Hello, I'm having some brain freeze on finding the derivative for this problem.

First, I changed the denominator to $\displaystyle {(1+12.6e^{-0.73x})^{-1}$, and then used the Product Rule. $\displaystyle f(x)=\frac{d}{dx}{[14x]}*{(1+12.6e^{-0.73x})^{-1}+\frac{d}{dx}{[(1+12.6e^{-0.73x})^{-1}]*{14x}$

That left me with: $\displaystyle f(x)={[14]}*{(1+12.6e^{-0.73x})^{-1}+{[(12.6e^{-0.73x})^{-1}]*{14x}$

This is where I'm getting brain frozen. I checked the back off the book it says the answer is: $\displaystyle f(x)={[14]}*{(1+12.6e^{-0.73x})^{-1}+ {128.772x(1+12.6e^{-0.73x})^{-2}e^{-0.73x}$.

I'm not sure where the value of $\displaystyle {128.772x(1+12.6e^{-0.73x})^{-2}e^{-0.73x}$ came from. I see parts of the Chain Rule, but the 128.772 has me dumbfounded. If there's a coefficient in front of $\displaystyle {e}$ what happens when taking the derivative of $\displaystyle {e}^{x}$?

Re: Derivative of 14x/(1+12.6e^(-0.73x)

Sorry about my handwriting

http://i.imgur.com/lEOEG.png

The last step I just used the distributive property

Re: Derivative of 14x/(1+12.6e^(-0.73x)

During your step here:

$\displaystyle f(x)={[14]}*{(1+12.6e^{-0.73x})^{-1}+{[(12.6e^{-0.73x})^{-1}]*{14x}$

You should show how you used the Chain Rule to see where you made the error, since you never finished the Chain rule on the second term's $\displaystyle [(12.6e^{-0.73x})^{-1}]$

Re: Derivative of 14x/(1+12.6e^(-0.73x)

If you feel insistent about the Product rule:

http://i.imgur.com/XjbCy.png

I thought this might not be clear enough and I was going to make some changes but it's late and I have to sleep so bye for now

Re: Derivative of 14x/(1+12.6e^(-0.73x)

Hello, AZach!

Quote:

$\displaystyle f(x)\:=\:\frac{14x}{(1+12.6e^{-0.73x})}$

$\displaystyle \text{Hello, I'm having some brain freeze on finding the derivative for this problem.}$

$\displaystyle \text{First, I changed the denominator to }(1+12.6e^{-0.73x})^{-1},\,\text{ and then used the Product Rule.}$

$\displaystyle f'(x)\:=\:\frac{d}{dx}(14x)\cdot(1+12.6e^{-0.73x})^{-1}+\frac{d}{dx}[(1+12.6e^{-0.73x})^{-1}]\cdot(14x)$

$\displaystyle \text{That left me with: }\:f'(x)\:=\:(14)\cdot{(1+12.6e^{-0.73x})^{-1}+\underbrace{[(12.6e^{-0.73x})^{-1}]}_{\text{This is wrong!}}\cdot(14x)$

The derivative of: .$\displaystyle g(x) \;=\;\left(1+12.6\:\!e^{-0.73x}\right)^{-1}$

. . . . . . . . . . .is: .$\displaystyle g'(x) \;=\;-\left(1 + 12.6\:\!e^{-0.73x}\right)^{-2}\cdot 12.6\:\!e^{-0.73x}\cdot (-0.73)$

. . . . . . . . . . . . . . . . . . $\displaystyle =\;9.198\:\!e^{-0.73x}\left(1 + 12.6\:\!e^{-0.73x}\right)^{-2}$

. . . . . . . . . . . . . . . . . . $\displaystyle =\;\frac{9.198}{e^{0.73x}\left(1 + 12.6\:\!e^{-0.73x}\right)^2} $

Unless you are *very* good at handling negative exponents,

. . I do *not* recommend switching to the Product Rule.

Re: Derivative of 14x/(1+12.6e^(-0.73x)

Quote:

Originally Posted by

**Soroban**

. . . . . . . . . . .is: .$\displaystyle g'(x) \;=\;-\left(1 + 12.6\:\!e^{-0.73x}\right)^{-2}\cdot 12.6\:\!e^{-0.73x}\cdot (-0.73)$

First off, thanks for the thorough responses, Daigo and Soroban. For this quoted part, I think this is where one of my main problems has been. I knew the derivative of $\displaystyle {e}^{x}$ was itself, so I ignorantly assumed that a coefficient in front of the 'x' variable wouldn't alter that. Looking at this from a function point of view, is $\displaystyle 12.6e^{-0.73x}$ and $\displaystyle {-0.73x}$ both considered inside functions of $\displaystyle (1+12.6e^{-0.73x})^{-1}$?

Re: Derivative of 14x/(1+12.6e^(-0.73x)

Quote:

Originally Posted by

**AZach** Looking at this from a function point of view, is $\displaystyle 12.6e^{-0.73x}$ and $\displaystyle {-0.73x}$ both considered inside functions of $\displaystyle (1+12.6e^{-0.73x})^{-1}$?

Absolutely - it's a glass-onion / russian doll situation!

Just in case a picture helps...

http://www.ballooncalculus.org/draw/diffChain/nest4.png

... where (key in spoiler) ...

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