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 ${\displaystyle f:\mathbb{R} \to \mathbb{R}}$ is given by ${\displaystyle f(x)=x^2}$ the the derivative is ${\displaystyle f'(x)=2x}$ . At ${\displaystyle x=5}$ the function value is ${\displaystyle f(5)=25}$ but ${\displaystyle f'(5)=10}$ is the linear transformation

${\displaystyle x\mapsto 10x}$ .

Another if ${\displaystyle F(x,y)=x^2+3y}$ at ${\displaystyle x=a,y=b}$ it differential is the gradient

${\displaystyle \left[\frac{\part F(a,b)}{\part x},\frac{\part F (a,b)}{\part y}\right]}$

and determines the linear tranformation

${\displaystyle \left[\frac{\part F(a,b)}{\part x},\frac{\part F(a,b)}{\part y}\right]:\R^2\to\R}$

given by

${\displaystyle (x,y)\mapsto\frac{\part F(a,b)}{\part x}x+\frac{\part F(a,b)}{\part y}y}$

For a vector function ${\displaystyle F:\R^n\to\R^m}$ let us ilustrate with another beispiel: Suppose that

${\displaystyle \begin{bmatrix}v\\w\end{bmatrix}\mapsto\begin{bmatrix}5v+w\\v^2\\-v+8w\end{bmatrix} }$

then

${\displaystyle \begin{bmatrix}5&1\\2v&0\\-1&8\end{bmatrix}}$

is the Jacobian. So at ${\displaystyle v=2,w=3}$ the differential is the map

${\displaystyle \begin{bmatrix}v\\w\end{bmatrix}\mapsto \begin{bmatrix}5&1\\4&0\\-1&8\end{bmatrix} \begin{bmatrix}v\\w\end{bmatrix} =\begin{bmatrix}5v+w\\4v\\-v+8w\end{bmatrix} }$