- #1
MiniSmSm
- 13
- 0
hi guys
I need The proof from definition of the derivative for exponential and log Fn
can somebody help me !
I need The proof from definition of the derivative for exponential and log Fn
can somebody help me !
[HELP] Derivative Of The Exponential
- Thread starter MiniSmSm
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- Derivative Exponential
In summary, the conversation discusses the proof for the definition of the derivative for exponential and logarithmic functions. The conversation begins with the user asking for help and another user welcoming them to the forum. They then discuss the definition of the exponential function, with one user asking for clarification on how ex is defined. The conversation then moves on to using a power series to define the function and how to prove its derivative. The user provides a starting point for the proof and directs the other user to a resource for further help. Finally, the conversation ends with the user thanking the other for the help and mentioning that they have not yet proved the fact that e^{x+y}= e^xe^y from the power series definition.
- #2
tiny-tim
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Welcome to PF!
Hi MiniSmSm! Welcome to PF!
Let's start with the exponential.
First, what definition of the exponential function are you using?
Hi MiniSmSm! Welcome to PF!
Let's start with the exponential.
First, what definition of the exponential function are you using?
- #3
MiniSmSm
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Welcome Tiny-tim
I use e^x or a^x
I use e^x or a^x
- #4
tiny-tim
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Hi MiniSmSm!
(try using the X2 tag just above the Reply box)
Yes, but how are you defining ex ?
(try using the X2 tag just above the Reply box
)Yes, but how are you defining ex ?
- #5
MiniSmSm
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I don understand !
I mean as a power series
I mean as a power series
- #6
tiny-tim
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MiniSmSm said:
You mean ∑0∞ xn/n! ?
ok, then just differentiate that …
what do you get?
- #7
MiniSmSm
- 13
- 0
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 !
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 !
- #8
Bohrok
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Looks like you want to start this way:
Let f(x) = ex
[tex]f'(x) = \lim_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h} = \lim_{h\rightarrow 0} \frac{e^{x + h} - e^x}{h}[/tex]
You might want to look here to get a start on the proof
https://www.physicsforums.com/showpost.php?p=2521922&postcount=9
Let f(x) = ex
[tex]f'(x) = \lim_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h} = \lim_{h\rightarrow 0} \frac{e^{x + h} - e^x}{h}[/tex]
You might want to look here to get a start on the proof
https://www.physicsforums.com/showpost.php?p=2521922&postcount=9
- #9
MiniSmSm
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thank you
this is what I wanted
this is what I wanted
- #10
HallsofIvy
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That proof requires the fact that [itex]e^{x+y}= e^xe^y[/itex]. Have you proved that from the power series definition? If so, how?
- #11
MiniSmSm
- 13
- 0
Actually I haven`t :(
how ?
how ?
Related to [HELP] Derivative Of The Exponential
What is the derivative of the exponential function?
The derivative of the exponential function is the same as the original function. In other words, the derivative of e^x is also e^x.
How do you find the derivative of the exponential function?
To find the derivative of the exponential function, you can use the power rule or the chain rule. The power rule states that the derivative of x^n is nx^(n-1), so for e^x, the derivative is e^x. The chain rule states that for a function f(g(x)), the derivative is f'(g(x))g'(x). Applying this to e^x, we get e^x * 1, which simplifies to just e^x.
What is the significance of the derivative of the exponential function?
The derivative of the exponential function is important in various fields such as physics, engineering, and economics. It is used to model growth and decay, as well as to solve differential equations.
How does the derivative of the exponential function relate to other functions?
The derivative of the exponential function is closely related to the derivative of logarithmic functions. In fact, the derivative of ln(x) is 1/x, which can be derived from the derivative of e^x. This relationship is known as the inverse property of logarithms and exponentials.
Can the derivative of the exponential function be negative?
No, the derivative of the exponential function cannot be negative. Since the derivative of e^x is always e^x, which is a positive value, the derivative of the exponential function will always be positive. This is because the exponential function is always increasing, never decreasing.
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