1、 一、导数的四则运算法则 ( ) uv uv ! ! = ( ) uv uv uv ! ! =+ 2 u uv uv vv ! ! “ #$ = %& (- 二、基本导数公式 ( ) 0 c ! = 1 xx = ( ) sin cos xx ! = ( ) cos sin xx ! =“ ( ) 2 tan sec xx ! = ( ) 2 cot csc xx ! =“ ( ) sec sec tan xxx ! =“ ( ) csc csc cot xxx ! =“ # ( ) xx ee ! = ( ) ln xx aaa ! = ln x ( ) ! = 1 x ( ) 1 log
2、ln x a xa ! = ( ) 2 1 arcsin 1 x x ! = “ ( ) 2 1 arccos 1 x x ! =“ “ ( ) 2 1 arctan 1 x x ! = + ( ) 2 1 arccot 1 x x ! =“ + ( ) 1 x ! = ( ) 1 2 x x ! = - 三、高阶导数的运算法则 （1） ( ) ( ) ( ) ( ) ( ) ( ) ( ) n nn ux vx ux vx = !“ #$（2） ( ) ( ) ( ) ( ) n n cu x cu x = !“ #$（3） ( ) ( ) ( ) ( ) n n n u ax b a u
3、ax b += + !“ #$（4） ( ) ( ) ( ) ( ) ( ) ( ) () 0 n n nk kk n k uxvx cu xv x ! = “= #$ %& - 四、基本初等函数的 n 阶导数公式 （1） ( ) ( ) ! n n xn = （2） ( ) ( ) n ax b n ax b ea e + = (3) ( ) ( ) ln n xx n aaa = (4) ( ) ( ) sin sin 2 n n ax b a ax b n ! “# += + + $ %& ( )* +, (5) ( ) ( ) cos cos 2 n n ax b a ax b n !
4、 “# += + + $ %& ( )* +,(6) ( ) ( ) ( ) 1 1! 1 n n n n an ax b ax b + ! “# =$ %& + ( +(7) ( ) ( ) ( ) ( ) ( ) 1 1! ln 1 n n n n an ax b ax b ! “! += ! #$ %& +- 五、微分公式与微分运算法则 ( ) 0 dc= ( ) 1 dx xd x = ( ) sin cos dxx d x = ( ) cos sin dx x d x = ( ) 2 tan sec dxx d x = ( ) 2 cot csc dx x d x = ( ) sec
5、 sec tan dxxx d x = ( ) csc csc cot dx xx d x = ( ) xx de ed x = ( ) ln xx da a a d x = ( ) 1 ln dxd x x = ( ) 1 log ln x a dd x xa = ( ) 2 1 arcsin 1 d x dx x = ( ) 2 1 arccos 1 d x dx x = ( ) 2 1 arctan 1 dxd x x = + ( ) 2 1 arccot 1 dxd x x = +六、微分运算法则 ( ) duv d ud v = ( ) dc u c d u = ( ) du v v
6、 d uu d v =+ 2 u vdu udv d vv ! “# = $% &七、基本积分公式 kdx kx c =+ 1 1 x xd x c + =+ + ln dx xc x =+ ln x x a adx c a =+ xx edx e c =+ cos sin xdx x c =+ sin cos xdx x c = + 2 2 1 sec tan cos dx xdx x c x =+ 2 2 1 csc cot sin xdx x c x = + 2 1 arctan 1 dx x c x =+ + 2 1 arcsin 1 dx x c x =+ tan ln cos xd
7、x x c = + cot ln sin xdx x c =+ sec ln sec tan xdx x x c =+ csc ln csc cot xdx x x c =+ 22 11 arctan x dx c ax a a =+ + 22 11 ln 2 xa dx c xa axa =+ + 22 1 arcsin x dx c a ax =+ 22 22 1 ln dx x x a c xa =+ - 八、下列常用凑微分公式 积分型 换元公式 ( ) ( ) ( ) 1 fa xbd x fa xbda xb a += + u ax b =+ ( ) ( ) ( ) 1 1 fxxd
8、 x fxdx = ux = ( ) ( ) ( ) 1 ln ln ln fxd xfx dx x = ln ux = ( ) ( ) ( ) xx xx feed x fede = x ue = ( ) ( ) ( ) 1 ln xx xx faad x fada a = x ua = ( ) ( ) ( ) sin cos sin sin fxx d xfx dx = sin ux = ( ) ( ) ( ) cos sin cos cos fxx d xfx dx = cos ux = ( ) ( ) ( ) 2 tan sec tan tan fxx d xfx dx = tan u
9、x = ( ) ( ) ( ) 2 cot csc cot cot fxx d xfx dx = cot ux = ( ) ( ) ( ) 2 1 arctan arc n arc n 1 f x dx f ta x d ta x x = + arctan ux = ( ) ( ) ( ) 2 1 arcsin arcsin arcsin 1 fxd xfx dx x = arcsin ux = - 九、分部积分法公式 形如 na x xedx ，令 n ux = ， ax dv e dx = 形如 sin n xx d x 令 n ux = ， sin dv xdx = 形如 cos n xx d x 令 n ux = ， cos dv xdx = 形如 arctan n xx d x ，令 arctan ux = ， n dv x dx = 形如 ln n xx d x ，令 ln ux = ， n dv x dx = 形如 sin ax ex d x ， cos ax ex d x 令 ,sin ,cos ax ue x x = 均可。
