# chain rule issue with derivative involving ln and e

• December 5th 2009, 06:09 PM
Archduke01
chain rule issue with derivative involving ln and e
y = ln [(e^x + e^-x) / 2] We need to find the derivative of this.

So I extract the 1/2 so it becomes

y = 1/2 ln (e^x + e^-x)

I'm not sure how the chain rule is applied here. The final answer is; (e^x - e^-x) / (e^x + e^-x)
• December 5th 2009, 06:14 PM
Jhevon
Quote:

Originally Posted by Archduke01
y = ln [(e^x + e^-x) / 2] We need to find the derivative of this.

So I extract the 1/2 so it becomes

y = 1/2 ln (e^x + e^-x)

I'm not sure how the chain rule is applied here. The final answer is; (e^x - e^-x) / (e^x + e^-x)

"extract" the 1/2?? And exactly what rule says we can do that?!

$y = \ln \frac {e^x + e^{-x}}2$

$= \ln (e^x + e^{-x}) - \ln 2$

Now differentiate using the rule: $\frac d{dx} \ln f(x) = \frac {f'(x)}{f(x)}$
• December 5th 2009, 06:16 PM
VonNemo19
Quote:

Originally Posted by Archduke01
y = ln [(e^x + e^-x) / 2] We need to find the derivative of this.

So I extract the 1/2 so it becomes

y = 1/2 ln (e^x + e^-x)

I'm not sure how the chain rule is applied here. The final answer is; (e^x - e^-x) / (e^x + e^-x)

You can't 'extract' the 2. It is associated with the Logarithm. You must be careful

$y=\ln(e^x-e^{-x})-\ln2$

Now

Let $u=e^x-e^{-x}$

Then

$y'=\frac{u'}{u}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\coth{x}$
• December 5th 2009, 06:19 PM
Jhevon
Quote:

Originally Posted by VonNemo19
You can't 'extract' the 2. It is associated with the Logarithm. You must be careful

$y=\ln(e^x-e^{-x})-\ln2$

Now

Let $u=e^x-e^{-x}$

Then

$y'=\frac{u'}{u}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\coth{x}$

It's $e^x + e^{-x}$

@OP: If we wanted hyperbolic functions for the answer, we could have replaced $\frac {e^x + e^{-x}}2$ with $\cosh x$ from the get go
• December 5th 2009, 06:22 PM
Archduke01
Quote:

Originally Posted by Jhevon
"extract" the 1/2?? And exactly what rule says we can do that?!

$y = \ln \frac {e^x + e^{-x}}2$

$= \ln (e^x + e^{-x}) - \ln 2$

Now differentiate using the rule: $\frac d{dx} \ln f(x) = \frac {f'(x)}{f(x)}$

if c is a constant then d ca/dx = c * d a/dx

But my badness, seems like I misunderstood the rule.

I don't understand the coth x cosh x stuff; I don't think we've learned that yet.
• December 5th 2009, 06:24 PM
VonNemo19
Quote:

Originally Posted by Jhevon
It's $e^x + e^{-x}$ Oops

@OP: If we wanted hyperbolic functions for the answer, we could have replaced $\frac {e^x + e^{-x}}2$ with $\cosh x$ from the get go I know this. I was trying to get a "Hey, what is coth,?" out of the OP

Thanks for the correction.
• December 5th 2009, 06:29 PM
Archduke01
$= \ln (e^x + e^{-x}) - \ln 2
$

how did ln 1/2 become - ln 2?
• December 5th 2009, 06:30 PM
VonNemo19
Quote:

Originally Posted by Archduke01
$= \ln (e^x + e^{-x}) - \ln 2$ $
$

how did ln 1/2 become - ln 2?

From the basic log rule $\ln\frac{a}{b}=\ln{a}-\ln{b}.$
• December 5th 2009, 06:35 PM
Archduke01
Quote:

Originally Posted by VonNemo19
From the basic log rule $\ln\frac{a}{b}=\ln{a}-\ln{b}.$

Thanks bromosapien.

$y'=\frac{u'}{u}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\coth{x}
$

Is this from applying the chain rule?

How did you guys get rid of the - ln 2? I assume you derived it, getting - 1/2 but at that point I'm stumped.
• December 5th 2009, 06:41 PM
Jhevon
Quote:

Originally Posted by Archduke01
Thanks bromosapien.

$y'=\frac{u'}{u}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\coth{x}
$

Is this from applying the chain rule?

How did you guys get rid of the - ln 2? I assume you derived it, getting - 1/2 but at that point I'm stumped.

ln(2) is a constant....
• December 5th 2009, 06:43 PM
Archduke01
Quote:

Originally Posted by Jhevon
ln(2) is a constant....

Ah yes of course, I was just -

Thanks for the help, gentlemen. Sorry for the whole back and forth.
• December 5th 2009, 06:44 PM
VonNemo19
Quote:

Originally Posted by Archduke01
Thanks bromosapien.

$y'=\frac{u'}{u}=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\coth{x}$ $
$

Is this from applying the chain rule?

How did you guys get rid of the - ln 2? I assume you derived it, getting - 1/2 but at that point I'm stumped.

$\ln2$ is a constant. Constants dissapear when differentiating.

Regarding the chain rule...

You know that $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ right?

Well, like I said before (if you forget about that nasty sign mistake that I made), let $u=e^x+e^{-x}$, then by the chain rule we have

$\frac{dy}{dx}=\frac{dy}{du}{\frac{du}{dx}}=\frac{1 }{u}\frac{du}{dx}=\frac{1}{e^x+e^{-x}}\cdot(e^x-e^{-x})$.

See what I'm sayin' brominator?