chain rule

Think of two connected gears. If the big gear turns twice as fast as the small one, and the small one turns three times as fast as the crank you're holding, then turning the crank spins the big gear at 2 times 3 = 6 times your hand speed. Rates of change multiply down a chain. The chain rule is exactly this idea, applied to functions that feed into one another.

When a function is built by composition — f applied to g(x), written f(g(x)) — the chain rule says (f(g(x)))' = f'(g(x)) * g'(x). In words: differentiate the outer function (leaving the inside alone), then multiply by the derivative of the inside. In Leibniz notation it reads beautifully as dy/dx = (dy/du)(du/dx), as if the du's cancel. For example, to differentiate (3x^2 + 1)^5: the outside is (something)^5 with derivative 5(something)^4, and the inside 3x^2 + 1 has derivative 6x, so the answer is 5(3x^2 + 1)^4 * 6x.

The chain rule is the workhorse that makes the other rules go far, because almost everything you differentiate in practice is secretly a composition. The classic slip is forgetting the 'times the derivative of the inside' factor — writing the derivative of sin(2x) as cos(2x) instead of the correct cos(2x) * 2. Whenever there's a function tucked inside another, that inner derivative must come along for the ride.