教学内容 和、差、积、商的求导法则 如果函数u(x),v(x)在点x处可导,则它们的和、差、积、 商(分母不为零)在点x处也可导,并且 (1)(x)±(x)='(x)±v(x) (2)[u(x)·v(x)=l'(x)v(x)+l(x)v(x) G)ru(x)yu(x)v(x)-u(x)v(x) (v(x)≠0) 证(1)、(2)略 证(3)设f(x)=-,(v(x)≠0 u(x+h)u(x) f(x)=lim f(x+h-f(x) =lim v(x+) v(x) h h lim u(x+h)v(x)-=u(x)v(x+h) (x+h)v(xh lim Lu(x+h)-u(xlv()-u(x)Lv(x+h)-v()1 h→0 v(x+h)v(x)h u(x+h-u(x) v(x)-u(x) v(x+h)-v(x) =lm h h v(x+hv(x) l(x)(x)-u(x)(x) [v(x)2 f(x)在x处可导 推论 ()D∑(x)=∑f(x (2)[Cf(x)=C(x)2 教 学 内 容 一、和、差、积、商的求导法则 定理： 商 分母不为零 在点 处也可导 并且 如果函数 在点 处可导 则它们的和、差、积、 ( ) , ( ), ( ) , x u x v x x ( ( ) 0). ( ) ( ) ( ) ( ) ( ) ] ( ) ( ) (3)[ (2)[ ( ) ( )] ( ) ( ) ( ) ( ); (1)[ ( ) ( )] ( ) ( ); 2   −   =   =  +    =    v x v x u x v x u x v x v x u x u x v x u x v x u x v x u x v x u x v x 证(1)、(2)略. 证(3) , ( ( ) 0), ( ) ( ) ( ) = v x  v x u x 设 f x h f x h f x f x h ( ) ( ) ( ) lim 0 + −  = → h v x u x v x h u x h h ( ) ( ) ( ) ( ) lim 0 − + + = → v x h v x h u x h v x u x v x h h ( ) ( ) ( ) ( ) ( ) ( ) lim 0 + + − + = → v x h v x h u x h u x v x u x v x h v x h ( ) ( ) [ ( ) ( )] ( ) ( )[ ( ) ( )] lim 0 + + − − + − = → ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) lim 0 v x h v x h v x h v x v x u x h u x h u x h + + −  −  + − = → 2 [ ( )] ( ) ( ) ( ) ( ) v x u  x v x −u x v  x =  f (x)在x处可导. 推论： (1) [ ( )] ( ); 1 1   = =  =  n i i n i i f x f x (2) [Cf (x)] = Cf (x);