第二节函数的求导法则 、导数的四则运算法则 二、反函数的导数法则 三、复合函数的导数法则 ●四、初等函数的求导法则
第二节 函数的求导法则 一、导数的四则运算法则 二、反函数的导数法则 三、复合函数的导数法则 四、初等函数的求导法则
导数的四则运算法则 定理如果函数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)|u(x)±v(x)=u(x)±v(x); (2)|u(x):v(x=l(x)v(x)+ul(x)(x) (x)-l(x)v(x)-(x)(x) (v(x)≠0
一、导数的四则运算法则 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 2 (1) [ ( ) ( )] ( ) ( ); (2) [ ( ) ( )] ( ) ( ) ( ) ( ); ( ) ( ) ( ) ( ) ( ) (3) [ ] ( ( ) 0). ( ) ( ) u x v x u x v x u x v x u x v x u x v x u x u x v x u x v x v x v x v x = = + − =
证(1)、(2)略 证(3)设∫(x) u(r) ,(v(x)≠0), v f∫"(x)=li f(x+h-f(r) h→>0 u(x+h)u(x) =lim V(r+h) h lin u(x+h)v(x)-u(x)v(x+h) h→>0 v(x+h)v(e)h
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
=lim lu(r+h)-u(x)]v(x)-u(x)[v(+h)-v(x) v(+ hv(x)h u(x+h)-(x) v(x)-(x) v(x+h-v(x) =im h→0 v(x+hv(r) u(x)v(x)-u(xv(r) vr ∴∫(x)在x处可导
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)C∑f(x=∑fx) (2)[Cf(x=C(x); (3)ⅢIf(x)=f(x)/2(x)…fn(x) l: +…+f1(x)f2(x)…fm(x) =∑Ⅱf(x)/k(x) i=1k=1 k≠i
推论 (1) [ ( )] ( ); 1 1 = = = n i i n i f i 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