高数微积分公式大全
- 格式:pdf
- 大小:250.15 KB
- 文档页数:5
高等数学微积分公式大全
一、基本导数公式
⑴()0c ′= ⑵1
x x
µ
µµ−= ⑶()sin cos x x ′=
⑷()cos sin x x ′=− ⑸()2
tan sec x x ′= ⑹()2
cot csc x x ′=− ⑺()sec sec tan x x x ′=⋅ ⑻()csc csc cot x x x ′=−⋅ ⑼()x
x
e
e
′= ⑽()ln x
x
a
a
a ′= ⑾()1
ln x x
′=
⑿(
)1
log ln x
a
x a
′= ⒀()2
1arcsin 1x x
′=− ⒁()2
1arccos 1x x
′=−
−
⒂()21arctan 1x x ′=
+ ⒃()2
1arccot 1x x ′=−+⒄()1x ′=
⒅
1′=
二、导数的四则运算法则
()u v u v ′′′±=± ()uv u v uv ′′′=+ 2u u v uv v v ′′′− =
三、高阶导数的运算法则 (1)()()()
()
()
()()n n n u x v x u x v x ±=±
(2)()()
(
)
()n n cu x cu x =
(3)()()()
()n n n
u ax b a u
ax b +=+ (4)()()()
()()()()0
n
n n k k k n k u x v x c u x v x −=⋅=
∑
四、基本初等函数的n 阶导数公式 (1)()
()
!n n
x
n = (2)()()
n ax b n ax b e a e ++=⋅ (3)()
()
ln n x x n a a a =
(4)()()
sin sin 2n n
ax b a ax b n π
+=++⋅
(5) ()()
cos cos 2n n
ax b a ax b n π
+=++⋅
(6)()
()
()
1
1!
1n n n
n a n ax b ax b +⋅ =
− +
+ (7) ()()
()
()()
1
1!
ln 1n n n n
a n ax
b ax b −⋅−+=−
+
五、微分公式与微分运算法则 ⑴()0d c = ⑵()1
d x
x
dx µ
µµ−= ⑶()sin cos d x xdx =
⑷()cos sin d x xdx =− ⑸()2
tan sec d x xdx = ⑹()2
cot csc d x xdx =− ⑺()sec sec tan d x x xdx =⋅ ⑻()csc csc cot d x x xdx =−⋅ ⑼(
)x
x d e
e dx = ⑽()ln x x d a a adx = ⑾()1
ln d x dx x
=
⑿(
)1
log ln x
a
d dx x a =
⒀()21arcsin 1d x dx x =
− ⒁()21arccos 1d x dx x
=−− ⒂()2
1
arctan 1d x dx x
=
+ ⒃()21arccot 1d x dx x =−+ 六、微分运算法则
⑴()d u v du dv ±=± ⑵()d cu cdu = ⑶()d uv vdu udv =+ ⑷2
u vdu udv d v v − =
七、基本积分公式
⑴kdx kx c =+∫ ⑵1
1x x dx
c µµ
µ+=++∫ ⑶ln dx x c x
=+∫ ⑷ln x
x
a a dx c a
=+∫ ⑸x x e dx
e c =+∫ ⑹cos sin xdx x c =+∫ ⑺sin cos xdx x c =−+∫ ⑻2
21sec tan cos dx xdx x c x ==
+∫∫ ⑼2
21csc cot sin xdx x c x ==−+∫∫
⑽21arctan 1dx x c x =++∫ ⑾
arcsin dx x c + 八、补充积分公式
tan ln cos xdx 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
11arctan x
dx c a x a a
=++∫ 2211ln 2x a dx c x a a x a −=+−+∫
arcsin x c a + ln x =+