chain rule issue with derivative involving ln and e

**Archduke01** · December 5th 2009, 05:09 PM

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)

**Jhevon** · December 5th 2009, 05:14 PM

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: $\frac d{dx} \ln f(x) = \frac {f'(x)}{f(x)}$

**VonNemo19** · December 5th 2009, 05:16 PM

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

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

**Jhevon** · December 5th 2009, 05:19 PM

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

**Archduke01** · December 5th 2009, 05:22 PM

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.

**VonNemo19** · December 5th 2009, 05:24 PM

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.

**Archduke01** · December 5th 2009, 05:29 PM

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

how did ln 1/2 become - ln 2?

**VonNemo19** · December 5th 2009, 05:30 PM

Originally Posted by Archduke01

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

how did ln 1/2 become - ln 2?

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

**Archduke01** · December 5th 2009, 05:35 PM

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.

**Jhevon** · December 5th 2009, 05:41 PM

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....

**Archduke01** · December 5th 2009, 05:43 PM

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.

**VonNemo19** · December 5th 2009, 05:44 PM

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?

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