The chain rule is used for differentiating composite functions and connected rates of change.

\(y = f\big(g(x)\big) \)

\(\boxed{\dfrac{dy}{dx} = f'\big(g(x)\big)g'(x)} \)

Using substitution, if \(y = f(u) \) and \(u = g(x) \), then:

\(\boxed{\dfrac{dy}{dx} = \dfrac{dy}{du} × \dfrac{du}{dx}} \)

Tip: Think of it as: "the derivative of the 'outside' function, multiplied by the derivative of the 'inside' function".

The chain rule can be "chained" for more than two functions. For three functions, if \(y = f(u) \), \(u = g(v) \) and \(v = h(x)\), then:

\(\dfrac{dy}{dx} = \dfrac{dy}{du} × \dfrac{du}{dv} × \dfrac{dv}{dx} \)