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Directional derivative

The derivative of a single-variable function is defined as

Using the extension of limits discussed above, one can then extend the definition of the derivative to a scalar-valued function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along some path s (t): R → R n {\displaystyle s(t):\mathbb {R} \to \mathbb {R} ^{n}}:

Unlike limits, for which the value depends on the exact form of the path s (t) {\displaystyle s(t)}, it can be shown that the derivative along the path depends only on the tangent vector of the path at s (t 0) {\displaystyle s(t_{0})}, i.e. s ′ (t 0) {\displaystyle s'(t_{0})}, provided that f {\displaystyle f} is Lipschitz continuous at s (t 0) {\displaystyle s(t_{0})}, and that the limit exits for at least one such path.

It is therefore possible to generate the definition of the directional derivative as follows: The directional derivative of a scalar-valued function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } along the unit vector u ^ {\displaystyle {\hat {\mathbf {u}}}} at some point x 0 ∈ R n {\displaystyle x_{0}\in \mathbb {R} ^{n}} is

or, when expressed in terms of ordinary differentiation,

which is a well defined expression because f (x 0 + u ^ t) {\displaystyle f(x_{0}+{\hat {\mathbf {u}}}t)} is a scalar function with one variable in t {\displaystyle t}.

It is not possible to define a unique scalar derivative without a direction; it is clear for example that ∇ u ^ f (x 0) = − ∇ − u ^ f (x 0) {\displaystyle \nabla _{\hat {\mathbf {u}}}f(x_{0})=-\nabla _{-{\hat {\mathbf {u}}}}f(x_{0})}. It is also possible for directional derivatives to exist for some directions but not for others.