[HELP] Derivative Of The Exponential

[MiniSmSm]

hi guys

I need The proof from definition of the derivative for exponential and log Fn
can somebody help me !

 

[tiny-tim]

Welcome to PF!

Hi MiniSmSm! Welcome to PF!

Let's start with the exponential.

First, what definition of the exponential function are you using?

 

[MiniSmSm]

Welcome Tiny-tim

I use e^x or a^x

 

[tiny-tim]

Hi MiniSmSm!

(try using the X² tag just above the Reply box )

Yes, but how are you defining e^x ?

 

[MiniSmSm]

I don understand !
I mean as a power series

 

[tiny-tim]

I mean as a power series

You mean ∑₀^∞ xⁿ/n! ?

ok, then just differentiate that …

what do you get? ​

 

[MiniSmSm]

No no ..

I mean by using this formula

F(x+h)-F(x) / h ( sorry I don`t know how to use tags :( )

how can I prove that the derivative of that function equals to its own !

 

[Bohrok]

Looks like you want to start this way:

Let f(x) = e^x
$$f'(x) = \lim_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h} = \lim_{h\rightarrow 0} \frac{e^{x + h} - e^x}{h}$$

You might want to look here to get a start on the proof
https://www.physicsforums.com/showpost.php?p=2521922&postcount=9

 

[MiniSmSm]

thank you
this is what I wanted

 

[HallsofIvy]

That proof requires the fact that $e^{x+y}= e^xe^y$. Have you proved that from the power series definition? If so, how?

 

[MiniSmSm]

Actually I haven`t :(
how ?