Derivative at a point

In mathematics, the derivative of a function is a measure of how rapidly the function changes locally when its argument changes.

Formally, the derivative of the function f at a is the limit

$f'(a)=\lim _{h\to 0}{f(a+h)-f(a) \over h}$

of the difference quotient as h approaches zero, if this limit exists. If the limit exists, then f is differentiable at a.

Multivariable calculus

The extension of the concept of derivative to multivariable functions, or vector-valued functions of vector variables, may be achieved by considering the derivative as a linear approximation to a differentiable function. In the one variable case we can regard $x\mapsto f(a)+f'(a)(x-a)$ as a linear function of one variable which is a close approximation to the function $x\mapsto f(x)$ at the point $x=a$.

Let $F:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{m}$ be a function of n variables. We say that F is differentiable at a point $a\in \mathbf {R} ^{n}$ if there is a linear function $\mathrm {D} F:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{m}$ such that

${\frac {\Vert F(a+h)-F(a)-\mathrm {D} F(h)\Vert }{\Vert h\Vert }}\rightarrow 0{\hbox{ as }}\Vert h\Vert \rightarrow 0\,$

where $\Vert \cdot \Vert$ denotes the Euclidean distance in $\mathbf {R} ^{n}$.

The derivative $\mathrm {D} F$, if it exists, is a linear map and hence may be represented by a matrix. The entries in the matrix are the partial derivatives of the component functions of F_j with respect to the coordinates x_i. If F is differentiable at a point then the partial derivatives all exist at that point, but the converse does not hold in general.