Chain rule (3D)
About this app
Two composed functions in 3D: x → u = g(x) → y = f(u). Slope triangles at P and Q share the common Δu — the teal pulsing "hinge" stroke makes visible why the derivatives multiply. Four combinations to choose from.
The idea
The chain rule tells you how to differentiate a composite function: (f ∘ g)′(x) = f′(g(x)) · g′(x). The inner function g sends x to u = g(x), and the outer function f sends u to y = f(u). When x changes by Δx, u changes by roughly g′(x)·Δx and y by roughly f′(u)·Δu. The rates of the two stages therefore multiply: Δy/Δx = (Δy/Δu) · (Δu/Δx).
What the app shows
A 3D scene places the two stages x → u and u → y side by side. Each curve carries a slope triangle, and both triangles share the same Δu, highlighted as a pulsing “hinge” segment. Four examples are available: sin(x²), √(x² + 1), sin²(x) and e^(−x²). The values panel lists g(x₀), f(g(x₀)), g′(x₀) and f′(g(x₀)) and checks the product f′(g(x₀))·g′(x₀) against the directly computed derivative (f ∘ g)′(x₀).
Things to try
Move x₀ and watch both factors change while their product keeps matching the direct derivative. The Δx slider makes the slope triangles larger or smaller; the smaller Δx, the closer the difference quotients get to the derivatives. The “Composition” toggle shows the curve of f ∘ g itself. With e^(−x²), notice how g′(x) = −2x decides the sign of the overall slope.
Frequently asked questions
- What is the chain rule?
- For a composite function h(x) = f(g(x)), the derivative is h′(x) = f′(g(x)) · g′(x). You differentiate the outer function, evaluate it at g(x), and multiply by the derivative of the inner function. In short: outer derivative times inner derivative.
- What is the derivative of sin(x²)?
- The inner function is g(x) = x² with g′(x) = 2x, the outer function is f(u) = sin u with f′(u) = cos u. By the chain rule the derivative is 2x·cos(x²). Forgetting the factor 2x is the most common mistake.
- Why do you multiply derivatives in the chain rule?
- A small change Δx produces a change Δu ≈ g′(x)·Δx in the inner function, which in turn produces Δy ≈ f′(u)·Δu. Substituting gives Δy ≈ f′(u)·g′(x)·Δx. The rates of the two stages multiply, much like two gears in series.
- What can I adjust in the chain rule app?
- You pick one of four examples, move the point x₀ and the step size Δx, and toggle the composite curve and the labels. A panel shows every intermediate value and compares the product of the derivatives with the direct derivative.
Subject: Analysis (one-dimensional)
More from Analysis (one-dimensional)
- Derivative as a sequence of secants
- Bounded growth
- Units of an integral: rate times time
- Exponential growth and decay
- Optimisation: the largest rectangle
- Area between curves
- Area function is an antiderivative
- Sequential continuity
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