Increase Functions: e^x and lnx

[Firepanda]

http://img135.imageshack.us/img135/9429/asdaauh6.jpg

I'm not too sure about this, I thought as a example function e^x, where I can say as x gets larger then so does e^x and then so does its derivative at those values of x.

But what if it was a function where as x increased the function was getting smaller, and so it would be a decreasing function.. or perhaps it doesn't matter and I should assume a function like that doesn't exist?

Is there any other function with that property other than e^x?

Really confused about the second part, taking e^x as an example, then then inverse would be lnx, and its derivative 1/x. Then surely I'm supposed to assume all they are talking about here is e^x, since no other f'(x) can take the form of 1/x other than f(x) = lnx...

 

[f(x)]

since f(x)=f`(x) and codomain is R+ f`(x) is always > 0 hence increasing.

 

[Firepanda]

since f(x)=f`(x) and codomain is R+ f`(x) is always > 0 hence increasing.

Would that be considered a proof?

 

[HallsofIvy]

What f(x) said is not quite true. f is NOT into R+ and the f ' is NOT always "> 0" but close:
Your problem tells you that f(x) is never negative and that f'(x)= f(x) for all x. That is, f'(x) is never negative. What does that tell you?

 

[Firepanda]

Ah I see, the second part though, so far I can say the slope of the inverse function would be given by 1/f'(x).

Can someone tell me if the question is asking for the derivative of the inverse function, or the inverse of the derivative?

Edit:

I see it's asking for the derivative of the inverse function, so what I did was this:

the slope of the inverse would be 1/f'(x) = dx/dy taking y = f(x)

then I went on to say that 1/f'(x) = (f -1)' (x) (derivative the inverse of f(x))

So howdo i show this is 1/x ?

 

[Firepanda]

gta hand this in in 30 mins, hoping someone will read this and help :(

 

[HallsofIvy]

You are confusing x and y in y= f(x) and x= f^-1(y).

 

[f(x)]

What f(x) said is not quite true. f is NOT into R+ and the f ' is NOT always "> 0" but close:
Your problem tells you that f(x) is never negative and that f'(x)= f(x) for all x. That is, f'(x) is never negative. What does that tell you?

Hello sir,
Pardon me but could you please explain why f`(x) >= 0 and not f`(x) > 0 since the codomain of f(x) has an interval (0,∞) exclusive while f(x) = f`(x)
Thanks

 

[HallsofIvy]

You are right, I was wrong. I misread $(0, \infty)$ as $[0, \infty)$.