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The chain rule states that the derivative of f (g (x)) is f' (g (x))⋅g' (x). In other words, it helps us differentiate *composite functions*. For example, sin (x²) is a composite function because it …

The chain rule can apply to composing multiple functions, not just two. For example, suppose A (x) , B (x) , C (x) and D (x) are four different functions, and define f to be their composition:

If we write f of x being equal to v of u of x, then we see very clearly the chain rule is very useful here. The chain rule tells us that f prime of x is going to be the derivative of v, with respect to u.

Chain rule Learn Chain rule Worked example: Derivative of cos³ (x) using the chain rule

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Chain rule Learn Chain rule Worked example: Derivative of cos³ (x) using the chain rule

Well we just said u is equal to sine of x, you reverse substitute, and you're going to get exactly that right over here. So when we talk about the reverse chain rule, it's essentially just doing u …

The chain rule for derivatives can be extended to higher dimensions. Here we see what that looks like in the relatively simple case where the composition is a single-variable function.

We explore the connection between the quotient rule, product rule, and chain rule in calculus. Rather than memorizing another rule, we see how the quotient rule naturally emerges from …