In mathematics , the directional derivative of a function that is dependent on several variables is the instantaneous rate of change of this function in a direction given by a vector.
The Gâteaux differential is a generalization of the directional derivation to infinitely dimensional spaces .
The directional derivative of a function at the point in the direction of is defined by the limit
if this exists.
Alternative definition
A function in the vicinity of 0 is defined by. is chosen so that the following applies
.
It is and the derivative of at the point is just the directional derivative of the point in the direction :
Unilateral directional derivatives
The unilateral directional derivatives of towards are defined by
The directional derivative in direction exists exactly when the two unilateral directional derivatives and coincide. In this case
Derivation in normalized directions
Some authors define the directional derivative only in the direction of normalized vectors:
These two definitions are the same for directions on the unit sphere . Otherwise the two definitions differ in terms of the factor . While the above definition is defined for all directions, the derivation in normalized directions is only defined for.
In applications in particular, it can be useful to calculate with the standardized direction vector ; this ensures that the directional derivative only depends on the direction, but not on the amount of .
In the one-dimensional case there are only two possible directions, namely to the left and to the right. The directional derivatives thus correspond to the usual one-sided derivatives. The derivatives in both directions can assume different values, which clearly means that the function can have a kink. A simple example of this is the amount function . It is not differentiable in, but the one-sided directional derivation exists:
For
and
For
The absolute amount is therefore equal to its one-sided directional derivative in 0 as a function of .