Rates Multiply, So the Inside Always Leaves a Factor

You'll learnto differentiate a composition by peeling it one layer at a time, and to see the inner factor that the most common error in all of calculus quietly drops.

A composition is a machine feeding a machine. Nudge x; the inside moves some multiple of that nudge; the outside responds to whatever arrives at its door. So the two rates multiply — dy/dx = dy/du · du/dx — and the derivative of sin(3x) is cos(3x) times 3, never cos(3x) on its own. That second factor is the entire content of the chain rule. It is also the single most commonly dropped object in differential calculus, and it survives because it is invisible in exactly one case: when the inside has derivative 1, multiplying by it changes nothing. Every other time it changes the answer.

8 STEPS · 6 QUICK CHECKS · dy/dx = dy/du · du/dx · COUNT THE MACHINES, THEN COUNT THE FACTORS · v1

y = f(g(x)) ⇄ dy/dx = f′(g(x)) · g′(x)

Before you start

What you're looking at

A composition drawn the way it actually works: two machines wired in series. x goes into the inner one, its output u goes into the outer one, and y comes out the far end.

The question

If the inside runs three times as fast as x, and the outside responds at its own rate per unit of u, how fast does y move per unit of x?

Watch for

One factor per machine — and step 7, where dropping the inner factor draws a tangent that visibly does not touch the curve.

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