The chain rule

Functions inside functions

Real models aren’t one simple function. They’re chains: the input goes through one function, whose output goes into another, and another. So we need to answer: if $y = f(g(x))$, how fast does $y$ change when we nudge $x$?

Think of gears. If gear x turns gear u three times as fast, and u turns gear y half as fast as itself, then y turns $3 \times \tfrac{1}{2} = 1.5$ times as fast as x. The rates multiply.