微积分公式技巧解答

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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

--→∞⎧=⎪⎪+++⎪

=<⎨+++⎪∞>⎪⎪⎩

（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）lim arctan 2x x π→∞= （6）lim tan 2

x arc x π

→-∞=-

（7）lim arc cot 0x x →∞

= （8）lim arc cot 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 a r c s i n x x arctan x x 2

11c o s 2

x x -

()ln 1x x + 1x e x - 1l n 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 μμμ-= 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(

)arcsin x '= ⒁(

)arccos x '=⒂()21arctan 1x x '=

+ ⒃()2

1arc cot 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()

1

d 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()1ln d x dx x

=

⑿(

)1

log ln x

a

d dx x a

= ⒀(

)arcsin d x = ⒁(

)arccos d x =