Why does the chain rule multiply the derivatives?
When one function sits inside another, f(g(x)), a small change in x passes through two stages, and each stage scales it by its own rate. So the overall rate is the product of the two: d/dx f(g(x)) = f′(g(x)) · g′(x).
Rates compound
Think of three linked gears. If gear A turns 3 times for each turn of gear B, and gear B turns 2 times for each turn of gear C, then A turns 3 × 2 = 6 times for each turn of C. A composite function works the same way. Let u = g(x) and y = f(u). Nudge x a little: u changes by about g′(x) times that nudge, and y changes by about f′(u) times the change in u. Multiply and you get the chain rule.
In Leibniz notation it reads dy/dx = dy/du · du/dx. The du terms are not really fractions that cancel, but the notation is built so the rule looks that way, which makes it easy to remember.
Worked examples
Differentiate y = (3x² + 1)⁵. The inner function is u = 3x² + 1, so du/dx = 6x. The outer function is u⁵, so dy/du = 5u⁴. Multiply: dy/dx = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴. At x = 1 that is 30 · 4⁴ = 7,680.
The rule also drives related rates. A circular ripple's radius grows at 2 cm/s. Its area is A = πr², so dA/dt = 2πr · dr/dt. When r = 5 cm, dA/dt = 2π(5)(2) = 20π ≈ 62.8 cm² per second. The area changes with r, r changes with t, and the rates multiply.
Common mistakes
- Forgetting the inner derivative: d/dx sin(2x) is 2cos(2x), not cos(2x).
- Evaluating the outer derivative at x instead of at g(x): for sin(x²) it is cos(x²) · 2x, not cos(x) · 2x.
- Stopping a layer too soon. For e^(sin 3x) there are three layers, so the derivative is e^(sin 3x) · cos(3x) · 3.
- Mixing it up with the product rule. The product rule is for f(x) · g(x), two functions multiplied; the chain rule is for f(g(x)), one inside the other.
Running the chain rule backward gives u-substitution for integrals, which is why it matters again once you reach the fundamental theorem of calculus.
Watching the stretch
In Isotoon, ask "Why does the chain rule multiply the derivatives?" and you get a short narrated animation. A clear version shows a small nudge in x stretched once by the inner function and again by the outer one, with the two stretch factors multiplying. If the idea of a rate is still shaky, start with what a derivative is. See also AP Calculus and calculus.
Questions
- How do I know when to use the chain rule?
- Whenever something other than plain x sits inside a function: sin(2x), (x² + 1)³, e^(5x), √(1 − x). If you could name an inner function u, you need the chain rule.
- Can the chain rule have more than two layers?
- Yes. Each layer adds one more factor. Work from the outside in, taking each derivative at whatever is inside it, and multiply all the factors together.