Sigmoid Derivative (Vectorized, Numerically Stable)

Difficulty: Medium · Concept: Activation Functions — Historical / Saturating Functions

Backpropagation through a sigmoid layer needs $\sigma'(z)$, not just $\sigma(z)$. Sigmoid has an unusually clean derivative in terms of its own output: $\sigma'(z) = \sigma(z)\big(1 - \sigma(z)\big)$ — no separate derivation needed once you already have $\sigma(z)$.

Two things make this harder than Sigmoid Activation From Scratch: it operates on a list of values, not just one, and — unlike the original problem's moderate test range — it has to stay numerically correct at extreme magnitudes. Sigmoid Activation's solution notes that 1 / (1 + math.exp(-z)) overflows for z below roughly -709, since that requires computing math.exp(1e3)-scale numbers. This problem actually requires fixing that, not just mentioning it.

Your task: implement sigmoid_derivative(z_list), returning $\sigma'(z)$ for every z in z_list. Must not raise OverflowError even for very large-magnitude inputs (e.g. -1000 or 1000).

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