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# Derivative at a point

The **derivative at a point** is a fundamental mathematical notion of calculus and analysis.
Informally, it shows the rate of change exhibited by a function at a particular point.

Probably the best known example of the derivative is velocity: While the mean velocity is obtained by dividing the distance travelled by the time needed, the velocity at a particular moment — as shown by a speedometer — is the differential quotient of the distance travelled (as function of the time needed) at that moment, i.e., for practical purposes, the mean velocity in a very short time interval or, in mathematical terms, the limit to that these mean velocities converge.

The derivative at a point does not always exist.
If, for some function, the derivative exists at all (or "almost all") points
then the resulting function is called the *derivative* of the function.

**Remark:**

The *derivative at a point* is a value defined locally to the point considered,
while the *derivative* (of a function) is a global function whose values coincide with the values obtained by the *derivatives at points*.
This subtle difference is easier to expressed using a term that is only rarely used in English, the *differential quotient*:
The differential quotient (the limit of the difference quotients) is the locally defined value (the *derivative at the point*) that
coincides with the value of the globally defined *derivative* at that point.

## Definition

The derivative of a real or complex function at a point is a measure of how rapidly the function changes locally (near this point) when its argument changes.

In order to define the derivative, a *difference quotient* is constructed.

The derivative of the function *f* at *a* is defined as the limit

as *x* approaches *x _{0}* or, equivalently,

*h*approaches zero, if this limit exists. The quotients of which the limit is taken are called

*difference quotients*.

If the limit exists, then *f* is said to be **differentiable at a**.
If a function is differentiable in all the points in which it is defined, then it is said to be **differentiable**.

If a function is differentiable in a point, then it is also continuous in that point. The reverse is not true as the following example shows:

- The absolute value
*f*(*x*) = |*x*| is continuous in the point 0, but not differentiable at 0:

- (We see that this expression has the limit 1 when we approach zero from the right side, but the limit −1 when we approach from the left side. Hence, the function is not differentiable.)

### Some notational styles

These are all equivalent ways to denote the derivative of a function *f* in the point *x*.

## 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 as a linear function of one variable which is a close approximation to the function at the point .

Let be a function of *n* variables. We say that *F* is differentiable at a point if there is a linear function
such that

where denotes the Euclidean distance in .

The derivative , 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.

## Formal derivative

The derivative of the monomial *X*^{n} may be formally defined as and this extends to a linear map *D* on the polynomial ring over any ring *R*. Similarly we may define *D* on the ring of formal power series .

The map *D* is a *derivation*, that is, an *R*-linear map such that