# Thread: Simple exponential derivative question

1. ## Simple exponential derivative question

I know this is simple, but I'm blanking - I think it's time for bed.

I need to find the derivative or the following.

$\displaystyle (e^w + 1) / e^w$

I would think it would go something like this...

(e^w)(e^w) - (e^w + 1) (e^w)

But I don't think I should be doing that - and if I do, I'm not sure what the next step is.

2. ## Re: Simple exponential derivative question

I think I see where I messed up.. I didn't put (e^w)^2 in the denominator. I'm assuming that essentially cancels the expressions in the numerator, right?

That would leave the answer at...

-(1/e^w)

Is this correct, and if so, have I gone about it the right way?

3. ## Re: Simple exponential derivative question

Originally Posted by astuart
I know this is simple, but I'm blanking - I think it's time for bed.

I need to find the derivative or the following.

$\displaystyle (e^w + 1) / e^w$

I would think it would go something like this...

(e^w)(e^w) - (e^w + 1) (e^w)

But I don't think I should be doing that - and if I do, I'm not sure what the next step is.
It's easier if you write \displaystyle \displaystyle \begin{align*} \frac{e^w + 1}{e^w} = 1 + \frac{1}{e^w} = 1 + e^{-w} \end{align*}.

4. ## Re: Simple exponential derivative question

Originally Posted by Prove It
It's easier if you write \displaystyle \displaystyle \begin{align*} \frac{e^w + 1}{e^w} = 1 + \frac{1}{e^w} = 1 + e^{-w} \end{align*}.
I'm studying all of this via distance education, so sometimes it can take a little bit to get the idea to set in, so this'll probably be another silly question.

Where you've written $\displaystyle 1 + 1 / e^w$ how did you get to this point?

The only way I can see it being done, is similar to the manner I got it there (chain rule), but with the fact that it was (e^w)^2 in the denominator that makes it stay as that expression (not squared). When compared to the numerator, which only had (e^w), this can simply be represented by 1 - almost like factorization.

Any chance you could clear that up?

5. ## Re: Simple exponential derivative question

Originally Posted by astuart
I'm studying all of this via distance education, so sometimes it can take a little bit to get the idea to set in, so this'll probably be another silly question.

Where you've written $\displaystyle 1 + 1 / e^w$ how did you get to this point?

The only way I can see it being done, is similar to the manner I got it there (chain rule), but with the fact that it was (e^w)^2 in the denominator that makes it stay as that expression (not squared). When compared to the numerator, which only had (e^w), this can simply be represented by 1 - almost like factorization.
$\displaystyle \frac{e^x+1}{e^x}=\frac{e^x}{e^x}+\frac{1}{e^x}=1+ \frac{1}{e^x}=1+e^{-x}$

$\displaystyle D_x(1+e^{-x})=-e^{-x}=\frac{-1}{e^x}$