AP® Calculus AB · Unit 3 · Topic 3.1
The Chain Rule
The chain rule says that to differentiate a function nested inside another, you multiply the derivative of the outer function by the derivative of the inner function. Rates of change compound, so their derivatives multiply.
Think of a composite function as one process feeding another: f(g(x)) first runs x through g, then feeds that result into f. Each piece has its own rate of change, and those rates stack on top of each other. If g is changing x at some rate, and f is changing g(x) at some rate, then the overall rate at which f(g(x)) changes is the product of the two. Nothing adds here, because the effects are not side by side, they are one inside the other, so they multiply.
Picture two meshed gears. Turn the small gear and the large gear turns too, but at a rate scaled by the gear ratio. The small gear is the inner function, the large gear is the outer one, and the ratio between their speeds is exactly what the chain rule captures: d/dx[f(g(x))] = f'(g(x)) times g'(x). You differentiate the outer function first, leaving the inner function untouched inside it, then multiply by the derivative of that inner function. For sin(3x2 + 1), the outer function is sine, so you get cos(3x2 + 1), then multiply by the derivative of the inside, 6x, giving 6x cos(3x2 + 1). For (5x - 2)7, the outer function is the seventh power, so bring down the exponent: 7(5x - 2)6, then multiply by the derivative of the inside, 5, giving 35(5x - 2)6.
This same layering shows up again almost immediately. When an equation mixes x and y and you cannot solve for y, differentiating y2 or sin(y) with respect to x still requires the chain rule, because y is secretly a function of x. That is the seed of implicit differentiation: every time you differentiate a y-term, you multiply by dy/dx, the same inner-derivative move you are practicing here.
Key terms
- Composite function
- A function built by plugging one function into another, written f(g(x)).
- Outer function
- The last operation applied in a composite function, the one wrapped around everything else.
- Inner function
- The function nested inside another, evaluated first before the outer function acts on it.
- Chain rule
- The rule stating that d/dx[f(g(x))] equals f'(g(x)) multiplied by g'(x).
Watch it explained
- How it worksHow do meshed gears and the chain rule say the same thing about rates?
- Worked exampleWalk me through differentiating sin(3x2 + 1) with the chain rule.
- Worked exampleFind the derivative of (5x - 2)7 using the chain rule.
- Common mistakeWhy isn't the derivative of sin(3x) simply cos(3x)?
- Ties togetherHow does this same layer-by-layer multiplying reappear when you differentiate an equation implicitly two topics from now?
Check yourself
End of topic.