Chapter 3 12 12.Differentiation of Matrix Function SUM RULE If A()and B(t)are both differentiable matrices A(0)+B)=(A()+B(》 di Proof:Let a0)+b1(0… aim(1)+bim(t) C(t)=A()+B(t)= a21(t)+b21() a2m()+b2m() an(1)+b(t).. anm(I)+bnm(t)） d(an+bu) d(aim+bim) dC①= dr dt dt d(an+b) d(abm) then di dt (a1+b)… (am+bm) = (an+bn）… (ann+bnm)】 an =A(t)+B(t) PRODUCT RULE A(t)and B(t)are both differentiable matrices,then AOB》=A)B)+AB d 12. Differentiation of Matrix Function SUM RULE If and are both differentiable matrices A (t ) B (t ) ( (t ) (t ) ) ( (t ) (t ) ) d t d A B A B    Proof: Let                      ( ) ( ) ... ( ) ( ) ... ... ... ( ) ( ) ... ( ) ( ) ( ) ( ) ... ( ) ( ) ( ) ( ) ( ) 1 1 2 1 2 1 2 2 1 1 1 1 1 1 a t b t a t b t a t b t a t b t a t b t a t b t t t t n n n m n m m m m m C A B then ( ) ( ) ... ... ... ... ... ... ... ... ... ... ( ) ... ( ) ... ... ... ( ) ... ( ) ( ) ... ( ) ... ... ... ( ) ... ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t t b b b b a a a a a b a b a b a b dt d a b dt d a b dt d a b dt d a b dt d t n n m m n n m m n n n m n m m m n n n m n m m m A B C                                                                           PRODUCT RULE A(t) and B(t) are both differentiable matrices, then ( (t ) (t ) ) (t ) (t ) (t ) (t ) d t d A B A B A B     Chapter 3 12