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This question already has an answer here:

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 prove it)?

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marked as duplicate by Paramanand Singh calculus Feb 21 '18 at 5:59

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    $\begingroup$ This isn't true, but it's very close to true. $f(x) = A e^x$ works for any real $A$. Taking $A=1$ or $A=0$ gives the two examples you gave in your question. The appropriate proof for this fact depends on how you've defined the exponential function - some authors define $e^x$ as the solution to your property which satisfies $f(0) = 1$. $\endgroup$ – B. Mehta Feb 20 '18 at 19:40
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Let's do a "function research" for functions of the type $f(x) = f^{\prime}(x)$.

From $f(x) = f^{\prime}(x)$ it follows that $f$ should be infinitely often differentiable.

So, you may try to check whether there is a meaningful Taylor series around $x= 0$ describing such an $f$. So set $A = f(0)$ and let's try $A \neq 0$. So, you get

$$\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n = A \sum_{n=0}^{\infty}\frac{1}{n!}x^n = A \cdot E(x)$$.

Now you study $E(x)$ and find all the nice properties like $E(x+y) = E(x)E(y)$ etc. and realize that this function is uniquely determined and is usally written as $e^x$.

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Any constant times $e^x$ also has this property.

To see that these are the only examples, suppose we have a function $g(x)$ with $g'(x)=g(x)$. Let $h(x)=g(x)e^{-x}$. Note that $e^x$ is never $0$ so $h(x)$ is well-defined. We compute $$h'(x)=g'(x)e^{-x}-g(x)e^{-x}=0\implies h(x)=\text {constant}$$

and we are done.

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  • $\begingroup$ Very nice. +1${}{}{}{}{}{}{}{}{}$ $\endgroup$ – DonAntonio Feb 20 '18 at 19:44
  • $\begingroup$ While I love this proof, OP should note that this assumes we know $e^x$ is its own derivative, and shows that it is the only function which is its own derivative (up to constant multiplication). $\endgroup$ – B. Mehta Feb 20 '18 at 19:44
  • $\begingroup$ @B.Mehta : your concern is addressed in this answer math.stackexchange.com/a/1292586/72031 $\endgroup$ – Paramanand Singh Feb 21 '18 at 14:11
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Let $y=f(x)$ thus the condition

$$f’(x)=f(x)$$

is a differential equation and for the Theorem of Existence and Uniqueness an unique solution exists up to a constant.

Notably by separation of variables

$$\frac{dy}{dx}=y\implies \frac{dy}{y}=dx\implies \log y = x+c$$

Which shows that the unique solution is the inverse of $\log$ function.

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