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The section Inverses in higher mathematics here says definitely the latter, but is the former idea often used in practice? For example, does this statement of the chain rule for one variable need extra caveats for

[tex]\left ( f \circ g \right )' = \left ( f' \circ g \right ) \cdot g'[/tex]

to be true in general? Specifically, would it only be true if [itex]f[/itex] was restricted to the domain of [itex]g[/itex] (rather than the condition being merely that [itex]f[/itex] is defined on an interval of which the range of [itex]g[/itex] is a subset), and differentiable on the range of [itex]g[/itex]?