Total derivative
The chain rule for total derivatives

The chain rule has a particularly elegant statement in terms of total derivatives. It says that, for two functions f {\displaystyle f} and g {\displaystyle g}, the total derivative of the composite function f ∘ g {\displaystyle f\circ g} at a {\displaystyle a} satisfies

If the total derivatives of f {\displaystyle f} and g {\displaystyle g} are identified with their Jacobian matrices, then the composite on the right-hand side is simply matrix multiplication. This is enormously useful in applications, as it makes it possible to account for essentially arbitrary dependencies among the arguments of a composite function.
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