The chain rule in Leibniz notation

The “official” form of the circle diagram method is the chain rule written in Leibniz notation: in that form the rule becomes almost obvious, because the differentials cancel like fractions.

Property — Chain rule

If $y = f(u)$ and $u = g(x)$, then: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}.$ For chains of three or more functions ($y = f(u)$, $u = g(v)$, $v = h(x)$): $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dv}\cdot\frac{dv}{dx}.$ The "$du$" and "$dv$" cancel like fractions. This is the deep meaning of Leibniz notation: it makes the chain rule an almost obvious fact.

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Topics: Derivatives
Concepts: Composite derivative · Leibniz notation · Chain rule
Methods: Differentiation rules
Skills: Differentiating
People: Gottfried Leibniz