Multiplication and the product-rule gradient

Read the equation

The local sensitivity to x is the current y value, and the sensitivity to y is the current x value. The backward pass multiplies each by the incoming gradient and accumulates the result. This is the product rule applied to one node of a computational graph.

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

// Multiplication: z = x * y, dz/dx = y, dz/dy = x
inline GradNodePtr mul(GradNodePtr x, GradNodePtr y, TapeRecorder& tape) {
    auto z = std::make_shared<GradNode>(x->value * y->value);
    z->inputs = {x, y};
    z->backward_fn = [x, y, z]() {
        x->grad += z->grad * y->value;
        y->grad += z->grad * x->value;
    };
    return tape.record(z);
}

Multiplication: z = x * y, dz/dx = y, dz/dy = x

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