= Solution
The <Feigenbaum period-doubling map> is quadratic-critical at zero: $g'(0)=0$ and $g''(0)\ne0$. Differentiating the renormalization equation once gives $g'(x)=g'(g(\beta x))g'(\beta x)$. Differentiate again and set $x=0$:
$$
g''(0)=\beta\left[g''(1)g'(0)^2+g'(1)g''(0)\right]
=\beta g'(1)g''(0).
$$
Cancellation of the nonzero <second derivative> therefore gives
$$
\boxed{g'(1)=\beta^{-1}\simeq-2.503.}
$$
The quadratic critical-point property matters. The normalization and functional equation alone, without specifying the Feigenbaum solution class, would not justify cancelling $g''(0)$.
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