基本初等函数求导公式
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基本初等函数求导公式
(1) 0)(='C
(2) 1)(-='μμμx x (3) x x cos )(sin ='
(4) x x sin )(cos -=' (5) a a a x x ln )(=' (6) (e )e x x '=
(7) a x x a ln 1
)(log =' (8) x x 1)(ln =
',
函数的和、差、积、商的求导法则
设)(x u u =,)(x v v =都可导,则
(1) v u v u '±'='±)(
(2) u C Cu '=')((C 是常数) (3) v u v u uv '+'=')(
(4) 2v v u v u v u '-'='⎪⎭⎫ ⎝⎛
反函数求导法则 若函数)(y x ϕ=在某区间y I 内可导、单调且0)(≠'y ϕ,则它的反函数)(x f y =在对应
区间x I 内也可导,且
)(1)(y x f ϕ'=' 或 dy dx dx dy 1=
复合函数求导法则
设)(u f y =,而)(x u ϕ=且)(u f 及)(x ϕ都可导,则复合函数)]([x f y ϕ=的导数为
dy dy du dx du dx =或
常用积分公式表·例题和点评
⑴d k x kx c =+⎰ (k 为常数) ⑵11d (1)1
x x x c μμμμ+≠-=++⎰
特别,
211d x c x x =-+⎰, 3223x x c =+⎰, x c =⎰ ⑶1d ln ||x x c x =+⎰ ⑷d ln x x a a x c a
=+⎰, 特别,e d e x x x c =+⎰ ⑸sin d cos x x x c =-+⎰
⑹cos d sin x x x c =+⎰
⑺221d csc d cot sin x x x x c x ==-+⎰⎰ ⑻221d sec d tan cos x x x x c x ==+⎰⎰ ⑼
arcsin (0)x x c a a
=+>,特别,arcsin x x c =+⎰ ⑽2211d arctan (0)x x c a a a a x =+>+⎰,特别,21d arctan 1x x c x =++⎰ ⑾2211d ln (0)2a x x c a a a x a x +=+>--⎰ 或2211d ln (0)2x a x c a a x a x a -=+>+-⎰