处也不可导 10.设∫(x)(i,j=12,…,n)为同一区间上的可导函数,证明 f1(x)f2(x)…fn( f1(x)f2(x)…fn(x) d|/2(x)f2(x)…Jn(x) f(x)f2(x)…f(x) Jn(x)fn2(x)…fm(x) M(x) fn2(x) fm(x) 证根据行列式的定义 f1(x)f2(x)…fn(x) dl/2(x)f2(x)…f2(x) dx fn(x)fm2(x)…Jm(x) a(-1)(k-+(x)5(x)…,(x) ∑(-1)4/(x)2(x)…fm(x)+f(x)(x)…(x)+ +f4(x)2(x)…m(x) ∫1(x)∫l2(x)…∫ln(x)f(x)f2(x)…fn(x) f1(x)f2(x) f2, (x).2I(x)f2(x)..f'2n(x) f,(x) fm,(x) fm(x)f(x)fn2(x) f(x) f1(x)f2(x) f(x) f1(x)f2(x) f,(x) fn(x)f"n(x)…f"m(x) G1(x)f2(x)…fn(x) ∑|f(x)f2(x)…fm(x) Jn(x)fn(x)…mn(x)处也不可导。 10．设 fij (x)（i, j = 1,2,", n）为同一区间上的可导函数，证明 ∑= = ′ ′ ′ n k n n nn k k kn n n n nn n n f x f x f x f x f x f x f x f x f x f x f x f x f x f x f x f x f x f x dx d 1 1 2 1 2 11 12 1 1 2 21 22 2 11 12 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) " # # # " # # # " " # # # " " 。 证 根据行列式的定义 11 12 1 21 22 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n nn f x f x f x d f x f x f x dx f x f x f x " " # # # " 1 2 1 2 ( ) 1 2 ( 1) ( ) ( ) ( ) n n N k k k k k nk d f x f x f x dx = − ∑ " " 1 2 1 2 1 2 1 2 ( ) 1 2 1 2 1 2 ( 1) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )] n n n n N k k k k k nk k k nk k k nk f x f x f x f x f x f x f x f x f x = − ′ ′ + + + ′ ∑ " " " " " 11 12 1 21 22 2 1 2 ' ( ) ' ( ) ' ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n n n nn f x f x f x f x f x f x f x f x f x = + " " # # # " 11 12 1 21 22 2 1 2 ( ) ( ) ( ) ' ( ) ' ( ) ' ( ) ( ) ( ) ( ) n n n n nn f x f x f x f x f x f x f x f x f x + " " " # # # " 11 12 1 21 22 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ' ( ) ' ( ) ' ( ) n n n n nn f x f x f x f x f x f x f x f x f x + " " # # # " 11 12 1 1 2 1 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) n n k k kn k n n nn f x f x f x f x f x f x f x f x f x = = ∑ ′ ′ ′ " # # # " # # # " 。 69