d'(x)=lim △(x=if()=f( △ 而Φ(a)=f(a),Φ(b)=f(b) 故vxeb有(x)可导,且a(x)=,/(x)d=f(x) 注2对于变上限的复合函数有以下两个推论 推论1若fx)在[a,b上连续,o(x)在[a,b上可导, d co(x) f(t)dt=f[(x)·@(x) (被积函数代积分上限且积分上限对x求导) 证变上限函数”fM可看成Y=)=(x)复合而成 d ro(x) dydy du 则 f(tdt f(u). o'(x)=flo(x)]o'(x) au ax3 lim ( ) ( ) x f f x   → = = 而 ( ) ( ), ( ) ( ) a f a b f b + +  =  =   故 有 可导    x a b x [ , ], ( ) , 注2 对于变上限的复合函数有以下两个推论 0 ( ) ( ) lim x x x  → x   =   ( ) [ ( ) ] ( ) x a  = =   x f x dx f x 且  推论1 若ƒ(x)在[a, b]上连续, (x)在[a, b]上可导, 则 ( ) ( ) [ ( )] ( ) x a d f t dt f x x dx  =     (被积函数代积分上限且积分上限对x求导) ( ) ( ) ( ), ( ) , x a f t dt Y u u x  =  = 证 变上限函数 可看成 复合而成  ( ) ( ) x a d dY f t dt dx dx  = 则  =  f x x [ ( )] ( )   dY du du dx =  =  f u x ( ) ( ) 