Editing Chain rule (section)

== Further generalizations ==
All extensions of calculus have a chain rule. In most of these, the formula remains the same, though the meaning of that formula may be vastly different.

One generalization is to [[manifold]]s. In this situation, the chain rule represents the fact that the derivative of {{math|''f'' ∘ ''g''}} is the composite of the derivative of {{math|''f''}} and the derivative of {{math|''g''}}. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

The chain rule is also valid for [[Fréchet derivative]]s in [[Banach space]]s. The same formula holds as before.<ref>{{cite book |first=Ward |last=Cheney |author-link=Elliott Ward Cheney Jr. |title=Analysis for Applied Mathematics |location=New York |publisher=Springer |year=2001 |chapter=The Chain Rule and Mean Value Theorems |pages=121–125 |isbn=0-387-95279-9 }}</ref> This case and the previous one admit a simultaneous generalization to [[Banach manifold]]s.

In [[differential algebra]], the derivative is interpreted as a morphism of modules of [[Kähler differential]]s. A [[ring homomorphism]] of [[commutative ring]]s {{math|''f'' : ''R'' → ''S''}} determines a morphism of Kähler differentials {{math|''Df'' : Ω<sub>''R''</sub> → Ω<sub>''S''</sub>}} which sends an element {{math|''dr''}} to {{math|''d''(''f''(''r''))}}, the exterior differential of {{math|''f''(''r'')}}. The formula {{math|1=''D''(''f'' ∘ ''g'') = ''Df'' ∘ ''Dg''}} holds in this context as well.

The common feature of these examples is that they are expressions of the idea that the derivative is part of a [[functor]]. A functor is an operation on spaces and functions between them. It associates to each space a new space and to each function between two spaces a new function between the corresponding new spaces. In each of the above cases, the functor sends each space to its [[tangent bundle]] and it sends each function to its derivative. For example, in the manifold case, the derivative sends a {{math|''C''<sup>''r''</sup>}}-manifold to a {{math|''C''<sup>''r''−1</sup>}}-manifold (its tangent bundle) and a {{math|''C''<sup>''r''</sup>}}-function to its total derivative. There is one requirement for this to be a functor, namely that the derivative of a composite must be the composite of the derivatives. This is exactly the formula {{math|1=''D''(''f'' ∘ ''g'') = ''Df'' ∘ ''Dg''}}.

There are also chain rules in [[stochastic calculus]]. One of these, [[Itō's lemma]], expresses the composite of an Itō process (or more generally a [[semimartingale]]) ''dX''<sub>''t''</sub> with a twice-differentiable function ''f''. In Itō's lemma, the derivative of the composite function depends not only on ''dX''<sub>''t''</sub> and the derivative of ''f'' but also on the second derivative of ''f''. The dependence on the second derivative is a consequence of the non-zero [[quadratic variation]] of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way. This variant of the chain rule is not an example of a functor because the two functions being composed are of different types.