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Chain Rule for Vectors

Derivation Math

y=f(g(x)),∂y∂x=∂f∂g⋅∂g∂xy = f(g(x)), \quad \frac{\partial y}{\partial x} = \frac{\partial f}{\partial g} \cdot \frac{\partial g}{\partial x}

Shapes: (k × m) · (m × n) = (k × n).

Why it matters: Backprop is this, applied right-to-left, never materializing full Jacobians — autograd stores each op’s vector-Jacobian product instead.