Interactive guide
One nudge, resized twice.
Inner: u = g(x)
Outer: y = f(u)
curve tangent nudged point (true)
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- True Δy
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- Predicted Δy
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Shrink Δx all the way down with the second slider and watch "True Δy" and "Predicted Δy" converge. The chain rule's product, f′(u)×g′(x), is only ever exact in that limit, at any real (nonzero) step size it's an approximation, a very good one close in, a worse one far out.
A composition is a pipeline, not a blend
f(g(x)) means: first run x through g, get a number u, then run that number through f. Nothing about f ever sees x directly, it only ever sees u. So a nudge to x has to travel through g first, arriving at f already resized by however much g stretches or shrinks things right there.
The chain rule is just bookkeeping for that two-stage resizing: g′(x) is how much g resizes a nudge at x; f′(u) is how much f resizes a nudge at u. Do both resizings in sequence, multiply, done.
Reading the two panels
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1
Left panel: Δx becomes Δu
The tangent line's slope, g′(x), is the resizing factor for this step.
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2
Right panel: that same Δu becomes Δy
Same idea, different slope: f′(u). The input here is the output of the left panel, not x.
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3
Two resizings in a row multiply
Resize by 3, then by 2, and the combined effect is resizing by 6, not 5. Rates compose by multiplying, not adding.
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4
Curved functions only agree with this locally
The tangent-based prediction and the true curve-based result match closely for small Δx, and drift apart for large ones.
Both tangent lines here are the exact idea from the Derivatives & Integration page, just used twice in a row instead of once. And the shrinking Δx slider is the exact idea from the Taylor Series page's higher terms: a smaller step means less of the curve's bending gets to matter, which is the entire reason a first-order, straight-line approximation can work at all.
The jargons
Two functions, one after the other, and the arithmetic for how their rates combine.
Applying g first, then f to whatever g produced. Also written (f∘g)(x).
The one applied first, directly to x. Its output, u, is the only thing the outer function ever sees.
The one applied last, to g's output rather than to x directly.
d/dx[f(g(x))] = f′(g(x)) × g′(x): multiply the two local resizing factors together.
A small step away from the current x, used to see how much the output moves in response.
Predicting a nudge's effect using only the tangent line's slope, exact in the limit, approximate everywhere else.