In modern mathematics the differential of a function is the linear transformation associated to each point in the domain of the function. This linear tranformation is given by the derivative.

For example if is given by the the derivative is . At the function value is but is the linear transformation

.

Another if at it differential is the gradient

and determines the linear tranformation

given by

For a vector function let us ilustrate with another beispiel: Suppose that

then

is the Jacobian. So at the differential is the map