Exercise 2: Sigmoid's Derivative at x=0 vs. Far From Zero — Possible Solution
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STEP 1: COMPUTE THE DERIVATIVE AT x=0
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Given sigma(0) = 0.5 exactly, using this chapter's own closed form:

sigma'(0) = sigma(0) * (1 - sigma(0)) = 0.5 * (1 - 0.5) = 0.5 * 0.5
= 0.25

STEP 2: COMPARE TO THE DERIVATIVE FAR FROM ZERO
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At a large x (e.g. x=10), sigma(x) approaches very close to 1 -
specifically sigma(10) is approximately 0.99995. Using the same
formula:

sigma'(10) = 0.99995 * (1 - 0.99995) = 0.99995 * 0.00005 =
approximately 0.0000454

STEP 3: INTERPRET USING THIS CHAPTER'S OWN SIGN/MAGNITUDE TABLE
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0.25 (at x=0) is a comparatively large derivative value, meaning
the function is changing relatively steeply there - per this
chapter's own table, a larger |f'(x)| means the function is
changing more rapidly.

0.0000454 (at x=10) is an extremely small derivative value, meaning
the function is nearly flat there - it's barely changing anymore,
since sigma(x) is already very close to its maximum value of 1 and
has little room left to increase.

RESULT
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sigma'(0) = 0.25, the LARGEST possible value the sigmoid derivative
ever takes (its peak steepness), while far from zero (large positive
or large negative x) the derivative shrinks toward 0, since the
function flattens out near its own upper and lower limits (1 and 0
respectively). Sigmoid is steepest right in its own middle and flat
at its extremes.

WHY THIS WORKS AS AN ANSWER
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Both derivative values are computed using this chapter's own exact
closed-form formula rather than estimated, and the comparison is
explicitly grounded in this chapter's own sign/magnitude
interpretation table (large vs. small |f'(x)|) rather than just
describing the numbers without connecting them to what magnitude
actually means geometrically.