ultivariate ifferentiation and ptimization the purpose of this appendix is to review various aspects of multivariate differentiation and optimization we assume the reader is familiar with differentiating realvalued function multivariate differentiation for multivariate function that maps vector [ xn ]to real number ( ) is the derivative taken with respect the partial derivative with respect to xi denoted to xi while all other variables are held constant we can write all the partial derivatives neatly using the "scalar/vectorderivative notationf :scalar/vector( partial derivative xn this vector of partial derivatives is known as the gradient of at and is sometimes written as (xnextsuppose that is multivalued (vector-valuedfunction taking values in rm defined by (xx (xx = (xxn fm (xwe can compute each of the partial derivatives fi / and organize them neatly in "vector/vectorderivative notationf xfm xfm ( vector/vector: xfmn xn xn gradient