Directional derivative

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The limit (if it exists) of the rate of change along a specified direction, say LaTeX: h,

LaTeX: \lim_{t \rightarrow 0^+} \frac{f(x+th) - f(x)}{t}.

In particular, the LaTeX: i-th partial derivative is with LaTeX: h=e_i. (Note that the directional derivative, as defined above, depends on the scale: the derivative for LaTeX: kh is LaTeX: k times the derivative for LaTeX: h. To avoid scale dependence, some authors require LaTeX: \|h\| = 1.) Recently, some people have called this a B-derivative (or Bouligand derivative), and functions that have directional derivatives in all feasible directions are B-differentiable. Some require the convergence to be uniform.


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