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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:\R\to\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

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

and determines the linear tranformation

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

given by

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

For a vector function F:\R^n\to\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}

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