Derivative of e^(x) evaluated at e

[gbn_jio]

Homework Statement

Okay, so.. I'm confused about this problem, because I'm nearly certain that there's no "correct" answer from the options given.

Consider the function f(x)=e^(x). Which of the following is equal to f'(e)? Note that there may be more than one


2. Answer options
a)
Lim (e^(x+h))/h
h->0

b)
Lim (e^(x+h)-e^(e))/h
h->0

c)
Lim (e^(x+h)-e)/h
h->0

d)
Lim (e^(x+h)-1)/h
h->0

e)
Lim (e^(e)) * (e^(h)-1)/h
h->0

f)
Lim e * (e^(h)-1)/h
h->0

The Attempt at a Solution

Since f(x)=e^(x)
f'(x)=e^(x) as well, and f'(e)=e^e

From that, I'm pretty sure that options a-d are undefined, as you can't divide by 0.

However, for option e: I got e^(e) - 1 as the answer (which isn't equal to f'(e))
and for option f: I got e-1 as the answer.

I'm confuzzled. Any help would be greatly appreciated!

 

[Feldoh]

Well F is the right answer... Have you ever proved that

$$\frac{d}{dx}e^x = e^x$$ just using the difference quotient?

If you haven't with the limit laws we can re-write f as:

$$\lim_{\substack{h\rightarrow 0}}e * \lim_{\substack{h\rightarrow 0}}\frac{e^h-1}{h} = \lim_{\substack{h\rightarrow 0}}e * 1 = e$$

 

[turdferguson]

What's the definition of a derivative and what's special about e?

 

[gbn_jio]

Well F is the right answer... Have you ever proved that

$$\frac{d}{dx}e^x = e^x$$ just using the difference quotient?

If you haven't with the limit laws we can re-write f as:

$$\lim_{\substack{h\rightarrow 0}}e * \lim_{\substack{h\rightarrow 0}}\frac{e^h-1}{h} = \lim_{\substack{h\rightarrow 0}}e * 1 = e$$

Sorry, that's what I meant instead of e-1.

That part does equal e, but it has to equal e^(e) instead of just e, since f(x)=e^x, f '(x)= e^(x), so f '(e)=e^(e)

Going by the limit laws, wouldn't the answer be option E?
Since it's the same thing as F, but with e^e instead... sorry I can't make it pretty, I'm not used to this

Thanks for helping...

 

[Feldoh]

Sorry, that's what I meant instead of e-1.

That part does equal e, but it has to equal e^(e) instead of just e, since f(x)=e^x, f '(x)= e^(x), so f '(e)=e^(e)

Going by the limit laws, wouldn't the answer be option E?
Since it's the same thing as F, but with e^e instead... sorry I can't make it pretty, I'm not used to this

Thanks for helping...

Oh shoot I'm sorry -- I wasn't thinking. The answer is letter e, but you still arrive at that answer by almost the same process as I just posted... factor out e^e and the everything else goes to 1. Does that make sense?