Are there any other functions which satisfy $\frac d{dx}f^2(x)=f(2x)$ other than $f(x)=\sin(x)$ and $f(x)=x$?

Question

I have been practicing functional equations recently and decided to do a functional equation that involves derivatives:$$\text{Find all functions }\mathbb R\to\mathbb R\text{ such that }\dfrac d{dx}f^2(x)=f(2x)$$Now a solution that can be found right away is the identity function $I(x)=x$, and the other solution I was able to find was $\sin(x)$ (which is done through the sine addition formula):$$\dfrac d{dx}\sin^2(x)=\dfrac d{dx}\sin(x)\sin(x)\\=\sin(x)\cos(x)+\sin(x)\cos(x)\\=2\sin(x)\cos(x)=\sin(2x)$$however I have been unable to find other solutions, so my question is: Are there other solutions to the functional equation $\dfrac d{dx}f^2(x)$, or are $f(x)=x$ and $f(x)=\sin(x)$ the only solutions?