第章 导数与微分 高等数学少学时 证以(uv)=dv+uv'为例，设fr(x)=(x)( 了'()=imf+ay)-fy) (x+△x)=△w+u(x) △x (x+△x)=△v+(x) 4(c+△)vx+=)v(x) △-→0 △x lim (u()+△uy(x)+△)-x)y(x) △x→0 △x lim v(x)△w+u(x)△v+△u△y △x-→0 Ar △u =v(x)lim +u(x)lim △v △u 会之+lim lim△y △x-→0△X x-→0△x Ax→0△x △-→0 =a(x)r(x)+(x)加'(x) 北京邮电大学出版社 33 ( ) ( ) ( ) ( ) x u x x v x x u x v x x     +  + −  = →0 lim = u  (x )v (x ) + u (x )v  (x ) ( )( ) ( ) ( ) x u x u v x v u x v x x     + + −  = → ( ) ( ) lim0 x v x u u x v u v x       + + = → ( ) ( ) lim0 v xu xv u x xu v x x x x x      0  0  0  0 ( ) lim ( ) lim lim lim → → → → = + +  设f (x) = u(x) v(x ). ( ) ( ) ( ) x f x x f x f x x   + −  = →0 lim u ( x + x ) =  u + u ( x ) v ( x + x ) =  v + v ( x ) 证 以 (uv ) = u  v + u v 为例 , 