The　directional　derivative　concept　of　the　multivariate　function　is　expanded　from　first　order　to　high　order　in　this　paper.　After　getting　the　definition　of　second　order　directional　derivative　and　the　calculation　formula,　the　high　order　directional　derivative　has　been　given.　Applications　of　the　high　order　directional　derivative　are　proposed　as　following:　1)　A　general　way　to　expanding　simple　variable　function　characteristics　to　multivariate　function　is　presented.　2)　The　necessary　conditions,　necessary　and　sufficient　conditions　of　extreme　value　criterion　of　function　are　easily　obtained.　3)　The　geometrical　meaning　of　semi-positive　and　semi-negative　definite　is　explained　according　to　second-order　directional　derivatives.　4)　It　is　revealed　that　there　is　extreme　value　problem　of　the　function　when　the　matrix　of　the　linear　equations　is　positive　or　negative　definite.　5)　Taylor＇s　expansion　formula　of　the　multivariate　function　is　easily　deduced.