Sigmoid derivative

Read the equation

The logistic derivative is z(1−z), where z is the forward sigmoid output. Its maximum is one quarter at z=0.5, and it decreases toward zero in either saturated tail. The primitive applies this local sensitivity through the chain rule.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

    z->inputs = {x};
    z->backward_fn = [x, z, t]() {
        x->grad += z->grad * (1.0 - t * t);
    };
    return tape.record(z);
}

// Sigmoid: z = 1/(1+exp(-x)), dz/dx = z*(1-z)
inline GradNodePtr sigmoid(GradNodePtr x, TapeRecorder& tape) {
    double s = 1.0 / (1.0 + std::exp(-x->value));
    auto z = std::make_shared<GradNode>(s);
    z->inputs = {x};
    z->backward_fn = [x, z, s]() {
        x->grad += z->grad * s * (1.0 - s);
    };
    return tape.record(z);
}

Sigmoid: z = 1/(1+exp(-x)), dz/dx = z*(1-z)

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