Owl: A hiker's comfort depends on air pressure. Air pressure depends on altitude. Altitude depends on time as they climb. Ask: how fast is comfort changing with respect to time?
You need a chain of rates: outer sensitivity times how fast the inner input is moving. That's the chain rule — multiply rates through composition.
It isn't a trig-only trick. Any nested structure f(g(t)) wants f'(g(t)) · g'(t).
Picture nested machines on the board: time → altitude → pressure → comfort. If you forget the inner derivative, what did you just pretend?