Definition
The directional derivative of a scalar function
along a vector
is the function defined by the limit
If the function f is differentiable at x, then the directional derivative exists along any vector v, and one has
where the on the right denotes the gradient and is the dot product. At any point x, the directional derivative of f intuitively represents the rate of change in moving at a rate and direction given by v at the point x.
Some authors define the directional derivative to be with respect to the vector v after normalization, thus ignoring its magnitude. In this case, one has
or in case f is differentiable at x,
This definition has several disadvantages: it only applies when the norm of a vector is defined and the vector is not null. It is also incompatible with notation used elsewhere in mathematics, where the space of derivations in a derivation algebra is expected to be a vector space.
Read more about this topic: Directional Derivative
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