The chain rule

Rates in a chain

Suppose u changes three times as fast as x, and y changes twice as fast as u. How fast does y change compared with x?

A composition y=f(g(x)) is a chain: x feeds u=g(x), and u feeds y=f(u). Below, a small step \Delta x on the top line is carried down the chain. Watch how each link stretches or shrinks it.

Drag the ochre marker on the x line, or use the sliders.

Composition
Δu ÷ Δx
Δy ÷ Δu
Δy ÷ Δx
f′(u) · g′(x)

The chain rule

For any step with \Delta u\ne0, the stretch factors multiply exactly:

\frac{\Delta y}{\Delta x}=\frac{\Delta y}{\Delta u}\cdot\frac{\Delta u}{\Delta x}.

Letting \Delta x\to0 turns each factor into a derivative. That is the chain rule:

\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx},\qquad\text{or}\qquad \big(f(g(x))\big)'=f'(g(x))\cdot g'(x).

In words: differentiate the outside function, leave the inside alone, then multiply by the derivative of the inside. Which part of \sin(x^2) is the inside?

Solve it step by step

Take each composition apart into an outside and an inside function, then rebuild the derivative. When a decimal is needed, two decimal places is enough. The score counts steps you answer correctly on the first try.

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    Common mistakes

    Forgetting the inside derivative

    \frac{d}{dx}(3x+1)^4 is 12(3x+1)^3, not 4(3x+1)^3. The factor 3 is the inside link of the chain.

    Changing the inside

    The outside derivative is evaluated at the inside, unchanged: \frac{d}{dx}\sin(x^2)=\cos(x^2)\cdot2x, not \cos(2x).

    Missing a hidden chain

    \sin^3x means (\sin x)^3, and \sqrt{x^2+9} means (x^2+9)^{1/2}. Both are compositions.