The cosine function satisfies its double-angle formula
$$(f(x))^2 = \frac{f(2x)+1}{2}$$
Now I was wondering if there are other continuous or even smooth (or other "sufficiently well behaved") functions $f: \mathbb K \to \mathbb K$ (with $\mathbb K = \mathbb R$ or $\mathbb C$) satisfying this equation.
My first attempt to solve this functional equation was differentiating it and applying techniques from solving differential equations, but unfortunately the $2x$ in the argument excludes standard methdos. For completeness sake, the equation would be
$$2f(x) f'(x) = f'(2x).$$
(With $f(0) = 1$ or $-0.5$ from the first equation.)
Are there any other methods to tackle this problem?