Hard

Sigmoid Gradient (Backprop)

Hard

~15 min

code completion

Sigmoid Gradient

During backpropagation, we need the derivative of each activation function. For sigmoid:

\frac{d σ}{d z} = σ (z) \cdot (1 - σ (z))

This elegant formula means: given the forward-pass output $σ (z)$ , the gradient costs almost nothing to compute.

The sigmoid gradient is maximized at $z = 0$ (value: 0.25) and approaches 0 for large $∣ z ∣$ . This causes the vanishing gradient problem in deep networks — gradients shrink as they flow backward through many sigmoid layers.

Your task:

Implement sigmoid_gradient(z) that returns the derivative of sigmoid at each element of z.

Example Tests

z=0: maximum gradient = 0.25

Input: {"z":0}

Expected: 0.25

Large positive z: gradient near 0

Input: {"z":100}

Expected: 0

Symmetric: gradient same at +1 and -1

Input: {"z":1}

Expected: 0.19661

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