# Function of a Function Differential Equation [duplicate]

Is there any function, $$f(x)\neq x$$, for which $$f(f'(x))=f'(f(x))$$?

## marked as duplicate by YuiTo Cheng, Ak19, cmk, Shogun, metamorphyJun 23 at 18:53

For the real-valued function $$f(x)=e^x$$, $$f'(x)=e^x$$.

• Thanks. $f(x)=e^x$ works. What if the condition was $f(x) \neq f'(x) \neq x$? – Hussain-Alqatari Jun 19 at 8:21
• There are still many. – Ivan Neretin Jun 19 at 8:24

Why, many. $$f(x)={x^2\over2}$$ will do.

If you want to exclude the "trick" answers where $$f'(x)=x$$ or $$f(x)=f'(x)$$, go with $$f(x)={x^3\over9}$$. Other examples are still plenty, I believe.

Upd. The other examples seem less numerous than I initially believed, but anyway, $$f(x)=\dfrac{x^n}{n^{n-1}}$$ for any constant $$n$$ is good.