Editing Automatic differentiation (section)

===Vector arguments and functions===

Multivariate functions can be handled with the same efficiency and mechanisms as univariate functions by adopting a directional derivative operator. That is, if it is sufficient to compute <math>y' = \nabla f(x)\cdot x'</math>, the directional derivative <math>y' \in \R^m</math> of <math>f:\R^n\to\R^m</math> at <math>x \in \R^n</math> in the direction <math>x' \in \R^n</math> may be calculated as <math>(\langle y_1,y'_1\rangle, \ldots, \langle y_m,y'_m\rangle) = f(\langle x_1,x'_1\rangle, \ldots, \langle x_n,x'_n\rangle)</math> using the same arithmetic as above. If all the elements of <math>\nabla f</math> are desired, then <math>n</math> function evaluations are required. Note that in many optimization applications, the directional derivative is indeed sufficient.