Vector Derivative (15 points ) Let X1, T2 be scalar values and x E R2x1 be the concatenate of x, and X 2 .
Let J = 2(21 (a) (2 pts ) Calculate a J . (b) (3 pts) Calculate second-order derivative ar, or, 02 J (c) (5 pts )
Calculate a x (d) (5 pts) Calculate second-order derivative . .(Hint: you should get 2 x 2 matrix )

Step-by-step explanation

J = 1 ( x, x2 - 4)" 9 ) J = 1 (2, 1 - 4)2 Partially differentiating writ . JJ = 12[ Ex,x2 -9 ( 2X, 22 -0) = (kix2 - y) ( 2 x x ) d x , JJ = 2X X_ (x, x 2 - 4) b) Now partially differentiating above equation dx, ox, JJ = 22, x2 2 (21 x2-4) + (x x2-4) &. (2x,x12) dx,dx _ dx, ox , = 2X,X2. (X,2- 0) + (x,2x2-y) (2x,) v . . = 2x,3x2 + 2x, (2, X2 - 4) = 2x, [xin + xix2 -41 = 22, [20,x2 - y] Mus . ... 1. 1.

C) J = 1 (2, " x2 - 4)" & Partially differentiating egn writ x Ox DJ = 1. 2 (x,2x, - 4) dixi-x-y) da = (x2x2-4) (224x2+ 2, - 0) = (X, x, -4) (2x, 212 +x /). d) Differentiating above ean wit x7 1 1 dxox JJ = ( x , x , - 4 ) @ ( 2x, x, + x," ) + ( 2 x,x2+x,) x dx 2 ( xin , - y ) = ( Xi x - y)(2x,+ 2x2+ 226,) + (2x12(2+X?) ( 276 , * 2 + 2 , 2 - 0) :. = (x x, - y) (4x, +2#2) + (2X, X2 + X, )(2X,x,+X;") = ( xix, - y) (ux, +2x,) + (2x, x,+ xing2