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Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?)

I was drawing some polynomials and their derivatives and noted that sometimes it was almost like the inverse. This lead me to ask this question: is there a function whose derivative is the inverse of that function?

Well, I figured that at least some kind of answer can be found to be of the form $a x^b$.

Lets solve this:

$f(x) = ax^b, f'(x) = abx^{b-1}$. Then $$f \circ f'(x) = a^{b+1}b^bx^{(b-1)b}=x=a^b b x^{(b-1)b}=f' \circ f (x).$$

Thus $b(b-1) = 1 \iff b^2-b-1=0 \iff b = \phi \vee 1-\phi,\ \phi = \frac{1+\sqrt 5}{2}$

We also see that $ab^{b-1}=1$, because both the multipliers must be one. Thus we get $a= \frac{1}{b^{b-1}}$. If $b=\phi$, we get $a=\phi^{\phi-1}$. If $b=1-\phi, \ a=(1-\phi)^{\phi}$

Thus two functions that satisfy the condition are $\phi^{\phi-1} x^\phi$ and $ (1-\phi)^{\phi}x^{1-\phi}$.

I would like to know if there are more functions like these, and do these functions have any 'interesting' properties, like exponential function, apart from this one condition about inverse being the derivative?

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The first assertion is wrong. The functions equal to their derivative are the functions $x\mapsto c\mathrm e^x$, for some $c$. –  Did Nov 18 '12 at 12:24
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@did: Actually it's not c times e to the power of x. It's c times e to the power of cx –  Armen Tsirunyan Nov 18 '12 at 12:25
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@ArmenTsirunyan No. –  Did Nov 18 '12 at 12:26
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$$(ce^{cx})'=c^2e^{cx}\neq ce^{cx}\Longrightarrow$$ did is right. –  DonAntonio Nov 18 '12 at 12:28
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This question appeared some time ago at mathoverflow. There you can find some solutions of the problem which are completely different from the above. See here: mathoverflow.net/questions/34052/function-satisfying-f-1-f/… –  Christian Blatter Nov 18 '12 at 12:49
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1 Answer 1

When you set $f^{-1}(x)=f'(x)$ these functions have the property that $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. \tag{1} $$ from "Inverse functions and differentiation".

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see here for a worked out example... –  draks ... Apr 28 at 19:38
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