# Linked Questions

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 ...
1answer
62 views

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 ...
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 $1$. ...
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 ...
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 ...
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 ...
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 ...
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 ...
4answers
109 views

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 $x \in \... 4answers 266 views ### Solve for$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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