A composite function is one function tucked inside another — an "outside" function acting on the result of an "inside" function. Two examples you'll see constantly this year:
In function notation, a composite is written h(t) = f(g(t)), where g(t) is the inside function and f(u) is the outside function acting on whatever g(t) produces. The power rule alone can differentiate f or g separately — but not the two of them wired together. That's the job of the chain rule.
To differentiate a composite function, differentiate the outside function first — leaving whatever is inside completely untouched — then multiply by the derivative of the inside function:
In plain language: derivative of the outside, times derivative of the inside. One more fact you'll need before applying this to physics: the derivatives of sine and cosine.
Before trusting the chain rule on something you can't easily double-check (like a trig function), it's worth verifying it against a case you can check — by expanding the composite function directly and differentiating the ordinary way.
Find h'(t) for h(t) = (3t + 1)², using the chain rule, then verify the result by expanding first.
Expanding isn't an option for a trig composite — there's no algebra trick that unpacks sin(ωt + φ) into separate pieces. The chain rule is the only way in.
Find dx/dt for x(t) = A sin(ωt + φ), where A, ω, and φ are constants.
Here's the payoff, and it's a big one for later this year. Unit 7 will hand you the equation x(t) = A cos(ωt + φ) and ask you to recognize it as the solution to a specific differential equation. Right now, with nothing but the chain rule, you can prove it's true.
Given x(t) = A cos(ωt + φ), find v(t) = dx/dt and a(t) = d²x/dt², and compare a(t) to x(t).
Explore this relationship directly below — no matter how you set A, ω, and φ, the acceleration graph always comes out as an upside-down, rescaled copy of the position graph.
Adjust amplitude A, angular frequency ω, and phase φ for x(t) = A cos(ωt + φ). Each derivative below was found using the chain rule — watch how a(t) always looks like an upside-down, rescaled copy of x(t).
a(t)'s graph is x(t)'s graph, flipped upside down and rescaled by ω². That's exactly what d²x/dt² = −ω²x means — acceleration is always proportional to position, and always points back the opposite way.