The definitions of directional derivatives for various situations are given below.

The directional derivative provides a systematic way of finding these derivatives.

Another way to think about tangent vectors is as directional derivatives.

In this case, the directional derivative is a vector in R.

The directional derivative therefore looks at curves in the manifold instead of vectors.

However, directional derivatives exist and yield support functions of support sets.

It can be used to calculate directional derivatives of scalar functions or normal directions.

The directional derivative is a special case of the Gâteaux derivative.

Hence the directional derivative at x is bounded below by the strictly positive constant on the right hand side.

This is an extension of the directional derivative to an infinite dimensional vector space.

Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.

A vector may also be defined as a directional derivative: consider a function and a curve .

This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued.

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector.

Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector.

The first expression is the directional derivative of in the direction of the vector field .

There is however another generalization of directional derivatives which is canonical: the Lie derivative.

Many of the familiar properties of the ordinary derivative hold for the directional derivative.

Some authors define the directional derivative to be with respect to the vector v after normalization, thus ignoring its magnitude.

Directional derivatives can be also denoted by:

Another fruitful point of view is to define the differential directly as a kind of directional derivative:

The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions.

In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves.

The 'covariant derivative' is a generalization of the directional derivative from vector calculus.

It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.