When one function feeds another, their slopes multiply. Nudge the input and watch the change travel down the chain — this is backpropagation, stripped to its essentials.
The chain diagram above the plot shows each link's local derivative. Their product is the total.
Derivatives of nested functions multiply — and that fact is the entire mechanism of backpropagation.
When a function feeds into another — y = f(g(x)) — a change in x must travel through both to reach y. The chain rule says the sensitivities simply multiply.
Picture two gears. Turning the first makes the second turn 3× as fast; that second gear drives a third at 2×. Turn the first gear once and the last one spins 6 times — the ratios multiplied.
Derivatives are exactly these ratios: how much does the output move per unit of input? Press Nudge x and watch a small change in x produce a change in u, which produces a change in y — each link scaling the one before.
The chain strip shows x → u → y with each link's local derivative underneath. The app also computes a numeric check — the slope measured by actually nudging x by a tiny amount — and it always matches the product. The rule is not an approximation.
A neural network is a deep composition: input → layer 1 → layer 2 → … → loss. To learn, you need the derivative of the loss with respect to a weight buried near the start. The chain rule provides it by multiplying every local derivative along the path:
Backpropagation is nothing more than this product, computed efficiently from the output backwards so shared terms are reused.
Multiplying many numbers together is unstable. If each local derivative is less than 1, the product shrinks toward zero — the vanishing gradient. If each is greater than 1, it explodes.
Choose the sigmoid outer function and slide x outwards. Its derivative never exceeds 0.25, so stacking ten sigmoid layers scales the gradient by at most 0.25¹⁰ ≈ 0.00000095. That is precisely why ReLU replaced sigmoid in deep networks, and why LSTMs added a gated path that avoids repeated multiplication.
Nested functions multiply their slopes. That gives you the derivative of arbitrarily deep compositions — and because multiplication compounds, it also explains why gradients vanish or explode in deep networks.
When one function feeds another, their slopes multiply. Nudge the input and watch the change travel down the chain — this is backpropagation, stripped to its essentials.