Is My Understanding of Basic Derivatives Correct?

[ggcheck]

[SOLVED] basic derivative question

since d/dx(e^x) = e^x

does d/dx(e^[-x]) = e^[-x]

and d/dx(e^[x+1])= e^(x+1)

the answer to one of my homework problems is different from my friends and I think that it is because I am mistaken about the stuff I just posted

 

[cristo]

No; \frac{d}{dx}(e^{f(x)})=f'(x)e^{f(x)}. Since in your first case, f(x)=1. f'(x)=1 so it doesn't matter, but it does for the second.

 

[ggcheck]

hmmm, what is wrong with this:

d/dx(e^(x+1)) = d/dx(e^[x] * e^[1]) = e^[x] * d/dx(e) + e * d/dx(e^x) = e^x * (0) + e * e^[x]= e^(x+1)

 

[cristo]

hmmm, what is wrong with this:

d/dx(e^(x+1)) = d/dx(e^[x] * e^[1]) = e^[x] * d/dx(e) + e * d/dx(e^x) = e^x * (0) + e * e^[x]= e^(x+1)

Nothing; that's correct. Alternatively, using the notation in my previous post, we have that f(x)=x+1, and so f'(x)=1 => d/dx(e^{x+1})=e^{x+1}

 

[ggcheck]

I thought you said that the second one isn't true?

 

[ggcheck]

I am confused; is d/dx(e^(x+1)) = e^(x+1)

 

[cristo]

I thought you said that the second one isn't true?

I did; the second one being d/dx(e^{-x}).

 

[ggcheck]

how can I find the derivative of e^[-x]

sorry for the confusion

 

[ggcheck]

wait, I did this to it... does this work:

e^-x = 1/e^x = e^x(0) + (1)(e^x)

 

[ggcheck]

Read post #2. If you have problems understanding that, then feel free to ask anything specific. I can't help you if you don't read what I write, can I?

I read post #2, I'm not familiar with that notation... I have never seen it before.

 

[cristo]

wait, I did this to it... does this work:

e^-x = 1/e^x = e^x(0) + (1)(e^x)

No, you need to use the quotient rule to differentiate quotients, something you may not have done yet.

Instead, read post #2. What is f(x) in this case?

 

[ggcheck]

No, you need to use the quotient rule to differentiate quotients, something you may not have done yet.

Instead, read post #2. What is f(x) in this case?

f(x) = -x

right?

 

[cristo]

f(x) = -x

right?

Correct. So, what's f'(x)? Note that the prime here just means "derivative wrt x" so, f'(x)=d/dx(f(x))=d/dx(-x)

[NB: I deleted my post a few above, the one you quoted, as it seemed a little abrupt. Sorry about that.]

 

[ggcheck]

which is -1?

 

[cristo]

which is -1?

Yup, and so, using the formula in #2, what is the derivative of e^{-x}?

 

[ggcheck]

-e^{-x} ?

 

[cristo]

-e^{-x} ?

Correct!

 

[ggcheck]

ugh, I am screwing up somewhere here...

e^{-x} = 1/(e^x) if I apply the quotient rule... [e^x * (0) - (1) * e^x] / (e^x)^2

-(e^x)/(e^{2x})

EDIT: switched signs

 

[ggcheck]

nm, after I cancel I get the same thing...

thank you very much for your help

 

[ggcheck]

would you mind taking a look at the problem that started all of this madness?

 

[ggcheck]

d/dx[e^(x+1) + e^x]/(e+1)

the answer that I got was e^x

 

[cristo]

d/dx[e^(x+1) + e^x]/(e+1)

the answer that I got was e^x

Well, this is equal to \frac{1}{e+1}\frac{d}{dx}(e^{x+1}+e^x}). From above, we know that d/dx(e^x)=e^x, and d/dx(e^{x+1})=e^{x+1}, so we obtain \frac{e^x+e^{x+1}}{1+e}=\frac{e^x(1+e)}{1+e}=e^x

 

[ggcheck]

Thanks a lot, btw what software are you using to type that out?

 

[cristo]

It's latex that's installed on the forum. If you click on advanced reply (or quote a post) then click on the little \Sigma icon on the toolbar you will have a drop down menu to use. Alternatively, simply click on some of the maths to view the input commands.

 

[ggcheck]

Thanks again.

 

[cristo]

Thanks again.

You're welcome!