The Jacobian matrix of a function is a matrix of partial derivatives. If is a function that maps from -dimensional space to -dimensional space, then the Jacobian matrix contains the first-order partial derivatives of each output dimension with respect to each input dimension. In other words, each entry of the Jacobian matrix represents how each output component changes with respect to each input component.

This is useful for finding the derivative of a vector function (ie. a function that takes a vector as an input and returns a vector as it's output for example the softmax function)

Formally for function , the jacobian would take the form

For example lets consider the Jacobian matrix for the function with components

The Jacobian matrix would then take the form