Exercise 3: Why w2's Chain Is Shorter Than w1's — Possible Solution
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WHY dL/dw2's CHAIN HAS ONLY TWO LINKS
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w2 appears directly in the equation for z2 (z2 = w2*a1 + b2), which
in turn feeds directly into L. There is exactly one composed
function between w2 and the final loss: w2 affects z2, and z2
affects L. That's the entire path - two links, dL/dz2 and dz2/dw2.

WHY dL/dw1's CHAIN NEEDS FOUR LINKS
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w1 appears in the equation for z1 (z1 = w1*x + b1) - but z1 does
NOT feed directly into L. Instead, z1 feeds into a1 (through the
sigmoid), a1 feeds into z2, and only then does z2 feed into L. So
the path from w1 to L passes through three intermediate composed
functions (z1 -> a1 -> z2 -> L), requiring one chain-rule link for
each step along that path: dL/dz2, dz2/da1, da1/dz1, dz1/dw1 - four
links total.

THE GENERAL PATTERN
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The length of the chain rule needed for any parameter's gradient
corresponds directly to how many layers of composition sit between
that parameter and the final loss - equivalently, how "deep" or
how early in the network that parameter sits. A parameter in the
very last layer (like w2, b2 here) needs only a short chain, since
it's directly connected to the loss. A parameter further back
(like w1, b1 here) needs a longer chain, since its effect on the
loss has to be traced forward through every intermediate layer it
passes through before reaching the output.

RESULT
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In a real, much deeper neural network, this same pattern continues:
a weight in the very first layer needs a chain rule product with as
many links as the network has layers, since its influence on the
final loss has to be traced all the way through every layer in
between. This is exactly why backpropagation processes a network
layer by layer, working backward from the loss - it's systematically
building up these longer and longer chains one link at a time,
reusing the pieces already computed for later layers rather than
recomputing the entire chain from scratch for every single
parameter.

WHY THIS WORKS AS AN ANSWER
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The explanation identifies the specific structural reason (number
of intermediate composed functions on the path to the loss) rather
than just observing that one chain happens to be shorter, and
generalizes the pattern to explain why backpropagation processes a
real, larger network the way it does - working backward, layer by
layer - connecting this small example directly to the real-world
algorithm it's meant to illustrate.