Linked Questions

2
votes
3answers
971 views

Why is $e^x$ the only function that is its own derivative? [duplicate]

I've heard that $f(x) = Ae^x$ is only function (both elementary and non-elementary) that satisfies the property $f(x)=\frac{df(x)}{dx}$. Is this true (and if it's true, is there a definitive way to ...
3
votes
1answer
62 views

On proving that $f(x) = f'(x)\iff f(x) = e^x$. (Not aware of a possible duplicate.) [duplicate]

This is just a curious question, but is the following true? $$f(x) = f'(x)\iff f(x) = e^x.$$ I can prove that $\dfrac{\mathrm d}{\mathrm dx}\left(e^x\right) = e^x$ from using the the formula, $e^x ...
108
votes
9answers
16k views

Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

I was wondering the following. And I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function ...
51
votes
10answers
2k views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
20
votes
4answers
4k views

How can I prove that there is a function that is its own derivative? [closed]

How can I prove that a function that is its own derivative exists? And how can I prove that this function is of the form $a(b^x)$?
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
12
votes
7answers
2k views

How do we show that the function which is its own derivative is exponential?

In my calculus class, to show that $\frac{d}{dx}e^x=e^x$ we did something like this: $$\lim_{h \to 0} \frac{a^{x+h} - a^x}{h} = a^x \lim_{h \to 0} \frac{a^h-1} h,$$ and then we defined $e$ to be the ...
4
votes
5answers
422 views

Why does the Taylor expansion of $e^x$ satisfy exponential properties?

Suppose I knew nothing about the function $e^x$. If I wanted to find a power series that was its own derivative, I would logically start with the constant term, and first start by setting it to $1$. ...
3
votes
6answers
196 views

Intuitive explanation of $y' = y \implies y = Ce^x$

I understand why $f : \mathbb{R} \to \mathbb{R}$ with $f'(x) = f(x)$ and $f(0) = 1$ must be $f (x) = e^x$, but I don't really feel it is super intuitive. Intuitively, why would you expect such a ...
2
votes
4answers
101 views

How to prove that $C_1e^x$ is the unique solution to $f'=f$?

Is it possible to prove that $C_1e^x$ is the unique solution to $f'(x)=f(x)$? I have tried to suppose there exists $g'=g$ and $g(x)\neq C_1e^x$. But I cannot find any contradiction by myself. Any ...
1
vote
5answers
140 views

Is there a way to remember the limit definition for $e$?

I occasionally see proofs where the limit definition for $e$ pops up and I don't recognize it for some reason, every time! $$e = \lim_{n \to \infty} \left( 1 + \frac{1}{n}\right)^n$$ For one thing I ...
2
votes
5answers
490 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...
1
vote
4answers
108 views

Prove that if $f=f'$ then $f$ is monotone.

Suppose I do not know that $e^x$ solves the equation $$f'(x)=f(x),\;\;\;x\in\mathbb{R}.$$ I am just given this equation and want to see if $f$ is increasing. Is there a way to prove that $f$ is ...
8
votes
4answers
109 views

Prove that $f(x) = 0$ for all $x \in \mathbb{R}$ (Analysis)

Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) = f^{(4)}(x)$ with $f(0) = f’(0) = f’’(0) = f’’’(0) = 0.$ Prove $f(x) = 0$ for all $x \in \mathbb{R}$ My Attempts: Suppose $x \in \...
1
vote
4answers
266 views

Solve for $f(x)=f'(x)$ without previous knowledge.

Solve for $f(x)=f'(x)$ without previous knowledge. I know it is obviously $f(x)=e^x$, but could you prove this without knowing $\frac d {dx}e^x=e^x$? And does there exist a $g(x)=g'(x)$ but $g(x)\ne ...

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