The equation should be from the Positive Reals to the Reals. I've tried to find a solution. I quickly found that it couldn't be strictly increasing (or decreasing) via considering that fact that $f(2)=f(1)+\int_{1}^{2}f'(x)dx$ and so it would have to oscillate.
I of course considered sin(x) for this, as that gives an infinite number of values of $k>1$ where $f'(x)=f(x+k)$, specifically $k=\frac{\pi}{2}+2n\pi$.
I then realised that the oscillations would have to become more spread out over time, so I considired $f(x)=\sin(\ln(x))$.
Of course there's the issue that this results in $f'(x)$ being much smaller in general then $f(x)$. So I know that you'll have to have something on the lines of $f(x)=\frac{\sin(\ln(x))}{x}$ or $f(x)=\frac{\sin(\ln(x))}{e^x}$. But I can't figure out an example. Can anyone else?
