# Differential

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

$x\mapsto 10x$.

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

$[\frac{\partial F (a,b)}{\partial x},\frac{\partial F (a,b)}{\partial y}]$

and determines the linear tranformation

$[\frac{\partial F (a,b)}{\partial x},\frac{\partial F (a,b)}{\partial y}]:\mathbb{R}^2\to\mathbb{R}$

given by $(x,y)\mapsto \frac{\partial F (a,b)}{\partial x}x+\frac{\partial F (a,b)}{\partial y}y$

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

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

then

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

is the Jacobian. So at $v=2, w=3$ the differential is the map

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