十、函数的和、差、积、商的求导法则定理1（四则运算法则）若函数u(x)、v(x)在点x处可导,(1)[u(x)±v(x))' = u'(x)±v'(x);(2)[u(x)v(x)])' = u(x)v(x) + u(x)v(x):u'(x)v(x) -u(x)v'(x)[](3)(v(x) ± 0)727v(x)(4)[Cu(x)]' = Cu'(x)
一 、函数的和、差、积、商的求导法则 (4)[Cu(x)] Cu (x) 定理1（四则运算法则） 若函数u(x)、v(x)在点 x 处可导， (1)[u(x) v(x)] u(x) v(x); (2)[u(x)v(x)] u(x)v(x) u(x)v(x); ( ( ) 0). ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3) 2 v x v x u x v x u x v x v x u x
证(2)设 f(x) = u(x)v(x),f(x+h)- f(x)u(x + h)v(x + h)-u(x)v(x)f'(x) = limlimhhh-→0h→>0u(x + h)v(x + h-u(x + h)v(x)+u(x +h)v(x)D-u(x)v(x)= limhh-→>0u(x +h)/v(x + h) -v(x)v(x)[u(x + h) -u(x)]= lim+ limhhh->0h-→>0= v'(x)limu(x + h) + v(x)u'(x)h->0?: u(x)可导,: u(x)连续,: limu(x+ h)=u(x)h0=v'(x)u(x) + v(x)u'(x)
证(2) 设 f (x) u(x)v(x), h f x h f x f x h ( ) ( ) ( ) lim 0 h u x h v x h u x v x h ( ) ( ) ( ) ( ) lim 0 h u x h v x h u x h v x u x h v x u x v x h ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) lim 0 h u x h v x h v x h ( )[ ( ) ( )] lim 0 h v x u x h u x h ( )[ ( ) ( )] lim 0 ( )lim ( ) 0 v x u x h h v(x)u(x) v(x)u(x) v(x)u(x) u(x)可导，u(x)连续， lim ( ) ( ) 0 u x h u x h ？