Chain rule
Derivative of a composition equals the composition of derivatives.
Let and be open sets. If is differentiable at and is differentiable at , then the composition is differentiable at , and
In terms of Jacobian matrices, this is
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Derivative of a composition equals the composition of derivatives.
Let and be open sets. If is differentiable at and is differentiable at , then the composition is differentiable at , and
In terms of Jacobian matrices, this is
An open set in a topological space is a subset such that .
Let be open, let , and let . The map is differentiable at if there is a linear map such that
where is small enough that , and is the Euclidean norm.
The map , which is necessarily unique, is the Fréchet derivative . In standard coordinates it is represented by the Jacobian matrix.
A composition of functions is the function obtained by applying one function after another: if and are functions (so the codomain of matches the domain of ), then the composition is defined by
A Jacobian matrix of a map (with ) at a point is the matrix
provided these partial derivatives exist.
When is differentiable at , the Fréchet derivative is a linear map, and is the matrix that represents in the standard bases of and . For scalar-valued , is closely related (up to transpose conventions) to the gradient.