第二节函数的和、差、积 商的求导法则 四一、和、差、积、商的求导法则 巴二、例题分析 小结思考题
、和、差、积、商的求导法则 中定理如果函数x),(x在点处可导则它 们的和、差、积、商分母不为零在点x处也 可导,并且 (1)[u(x)±v(x)=u'(x)±v(x) 工工工 (2)[u(x)·v(x)Y=u'(x)v(x)+u(x)y(x); u(x), u(v(x-u(x)v(x) (3) (v(x)≠0) v2(x) 上页
一、和、差、积、商的求导法则 定理 可 导 并 且 们的和、差、积、商 分母不为零 在 点 处 也 如果函数 在 点 处可导 则 它 , ( ) ( ), ( ) , 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)设∫(x)= ux ,(v(x)≠0), vlX f(r)=lim/(x+1)-f(x) h→0 h u(x+h u(x) =li v(x+h )w(x) h→>0 h =lil u(x+hv()-u(xv(x+h h→>0 v(x+hv()h 上页
证(3) , ( ( ) 0), ( ) ( ) ( ) = v x v x u x 设 f x h f x h f x f x h ( ) ( ) ( ) lim 0 + − = → v x h v x h u x h v x u x v x h h ( ) ( ) ( ) ( ) ( ) ( ) lim 0 + + − + = → h v x u x v x h u x h h ( ) ( ) ( ) ( ) lim 0 − + + = → 证(1)、(2)略
lu(x+h)-u(x)lv(x)u(xlv(x+h)-v() h→>0 v(+hv(x)h u(x+h-u(r (x)-(x),"(+b)-u(x) =lim h h→0 v(x+hv(x) u(x)v(x)-u(x)v(x) v(x)l f(x)在x处可导 上页
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 中()②fx=∑fx; i=1 i=1 (2)|(x)=C(x); (3)ⅢIf(x)=f1(x)1(x)…fn(x) i=1 工工工 +…+f1(x)2(x)…f(x) =∑Ⅱf(x)k(x); i=1k=1 ≠i 上页
推论 (1) [ ( )] ( ); 1 1 = = = n i i n i fi x f x (2) [Cf (x)] = Cf (x); ( ) ( ); ( ) ( ) ( ) (3) [ ( )] ( ) ( ) ( ) 1 1 1 2 1 2 1 = + + = = = = n i n k i k i k n n n i i f x f x f x f x f x f x f x f x f x