有关高等数学计算过程中所涉及到的数学公式(集锦)
一、0
101101lim 0n n n m m x m a n m b a x a x a n m b x b x b n m
--→∞⎧=⎪⎪+++⎪
=<⎨+++⎪∞>⎪⎪⎩
L L (系数不为0的情况) 二、重要公式(1)0sin lim 1x x
x
→= (2)()1
0lim 1x x x e →+=
(3))1n a o >=
(4)1n = (5)limarctan 2
x x π
→∞
=
(6)lim tan 2
x arc x π
→-∞
=-
(7)limarccot 0x x →∞
= (8)lim arccot x x π→-∞
= (9)lim 0x
x e →-∞
=
(10)lim x x e →+∞
=∞ (11)0
lim 1x
x x +
→= 三、下列常用等价无穷小关系(0x →)
sin x x : tan x x : arcsin x x : arctan x x : 2
11cos 2
x x -:
()ln 1x x +: 1x e x -: 1ln x a x a -: ()11x x ∂
+-∂:
四、导数的四则运算法则
()u v u v '''±=± ()uv u v uv '''=+ 2u u v uv v v '''-⎛⎫= ⎪⎝⎭
五、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()1ln x x
'=
⑿(
)
1
log ln x
a
x a '=
⒀(
)arcsin x '= ⒁(
)arccos x '=
⒂()21arctan 1x x '=
+ ⒃()2
1arccot 1x x
'=-+⒄()1x '=
⒅
'
=
六、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 -=⋅=⎡⎤⎣⎦
∑ 七、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 -⋅-+=-⎡⎤⎣⎦
+
八、微分公式与微分运算法则
⑴()0d c = ⑵()
1d x x dx μμμ-= 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(
)arcsin d x = ⒁(
)arccos d x =
⒂()21arctan 1d x dx x =
+ ⒃()2
1
arccot 1d x dx x =-+ 九、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 -⎛⎫= ⎪⎝⎭
十、基本积分公式
⑴kdx kx c =+⎰ ⑵11x 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 =-+⎰
⑻
2
21sec tan cos dx xdx x c x ==+⎰⎰
