导数与微分常用公式(基础)

导数与微分常用公式(基础)

一。导数的定义；

1.导数的定义：
导数其实就是函数某点附近的 00 0 0 型极限
f′(x)=limh→0f(x+h)−f(x)h f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h
左导数
f′(x−)=limh→0−f(x+h)−f(x)h f ′ ( x − ) = lim h → 0 − f ( x + h ) − f ( x ) h
右导数
f′(x+)=limh→0+f(x+h)−f(x)h f ′ ( x + ) = lim h → 0 + f ( x + h ) − f ( x ) h

连续的定义：
函数在x0可导，(左导数=右导数)，则连续，反之不成立。

几何意义：切线的斜率

2.高阶导数：
莱布尼兹公式：
(uv)(n)=∑ni=0Cinu(i)v(n−i) ( u v ) ( n ) = ∑ i = 0 n C n i u ( i ) v ( n − i )

二项分布公式
重复n次，发生i次的概率
P(X=i)=Cinp(i)q(n−i) P ( X = i ) = C n i p ( i ) q ( n − i )

3.求导方法
四则运算
(u±v)′=u′±v′ ( u ± v ) ′ = u ′ ± v ′
(uv)′=u′v+uv′ ( u v ) ′ = u ′ v + u v ′

uv=u′v−uv′v2,(v≠0) u v = u ′ v − u v ′ v 2 , ( v ≠ 0 )

4.反函数的求导法则
函数 y=f(x) ==》 x=f(y)
反函数 y=f−1(x) y = f − 1 ( x )
反函数的导数 y′=[f−1(x)]′=1f′(y) y ′ = [ f − 1 ( x ) ] ′ = 1 f ′ ( y )

5.复合函数的导数
y=f[g(x)]
dydx=dydududx d y d x = d y d u d u d x
y′(x)=f′(u)g′(x) y ′ ( x ) = f ′ ( u ) g ′ ( x )

二.常用基本初等函数的导数公式

(C)’=0 (C为常数)
(xa)′=axa−1 ( x a ) ′ = a x a − 1
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(ex)′=ex ( e x ) ′ = e x
(ax)′=axlna ( a x ) ′ = a x l n a , (a>0 且 a!=1)
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(ln|x|)′=1x ( l n | x | ) ′ = 1 x
(loga|x|)′=1xlna ( l o g a | x | ) ′ = 1 x l n a , (a>0 且 a!=1)
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(sinx)’=cosx
(cosx)’=-sinx
(tanx)′=1cos2x=sec2x ( t a n x ) ′ = 1 c o s 2 x = s e c 2 x
(cotx)′=−1sin2x=csc2x ( c o t x ) ′ = − 1 s i n 2 x = c s c 2 x
(secx)′=secx∗tanx ( s e c x ) ′ = s e c x ∗ t a n x
( secx=1cosx s e c x = 1 c o s x )
(cscx)′=−cscx∗cotx ( c s c x ) ′ = − c s c x ∗ c o t x
( cscx=1sinx c s c x = 1 s i n x )

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(arcsinx)′=11−x2√ ( a r c s i n x ) ′ = 1 1 − x 2
(arccosx)′=−11−x2√ ( a r c c o s x ) ′ = − 1 1 − x 2
(arctanx)′=11+x2 ( a r c t a n x ) ′ = 1 1 + x 2
(arccotx)′=−11+x2 ( a r c c o t x ) ′ = − 1 1 + x 2

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三.常用高阶导数公式
(ex)(n)=ex ( e x ) ( n ) = e x
(sinx)(n)=sin(x+nπ2) ( s i n x ) ( n ) = s i n ( x + n π 2 )
(cosx)(n)=cos(x+nπ2) ( c o s x ) ( n ) = c o s ( x + n π 2 )
[ln(1+x)](n)=(−1)n−1∗(n−1)!(1+x)n [ l n ( 1 + x ) ] ( n ) = ( − 1 ) n − 1 ∗ ( n − 1 ) ! ( 1 + x ) n
(xa)n=a(a−1)...(a−n+1)xa−n ( x a ) n = a ( a − 1 ) . . . ( a − n + 1 ) x a − n

莱布尼兹公式：
(u∗v)(n)=∑nk=0Cknu(k)v(n−k) ( u ∗ v ) ( n ) = ∑ k = 0 n C n k u ( k ) v ( n − k )

二项分布公式
重复n次，发生k次的概率
P(X=k)=Cknp(k)q(n−k) P ( X = k ) = C n k p ( k ) q ( n − k )

四.利用微分做近似计算的常用公式
基本公式
f(x0+Δx)≈f(x0)+f′(x0)Δx f ( x 0 + Δ x ) ≈ f ( x 0 ) + f ′ ( x 0 ) Δ x
Δy≈f′(x0)Δx Δ y ≈ f ′ ( x 0 ) Δ x

常用公式
（ |x|→0 | x | → 0 时）
ex≈1+x e x ≈ 1 + x
ln(1+x)≈x l n ( 1 + x ) ≈ x
(1+x)a≈1+ax ( 1 + x ) a ≈ 1 + a x
sinx≈x s i n x ≈ x
tanx≈x t a n x ≈ x
arcsinx≈x a r c s i n x ≈ x
cosx≈1−x22 c o s x ≈ 1 − x 2 2

五.微分基本公式

d(C) = 0
d(xa)=axa−1dx d ( x a ) = a x a − 1 d x
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d(ex)=exdx d ( e x ) = e x d x
d(ax)=lna∗axdx d ( a x ) = l n a ∗ a x d x , (a>0 且 a!=1)
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d(ln|x|)=1xdx d ( l n | x | ) = 1 x d x
d(loga|x|)=1xlnadx d ( l o g a | x | ) = 1 x l n a d x , (a>0 且 a!=1)
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d(sinx)=cosx dx
d(cosx)=-sinx dx
d(tanx)=1cos2xdx=sec2xdx d ( t a n x ) = 1 c o s 2 x d x = s e c 2 x d x
d(cotx)=−1sin2xdx=csc2xdx d ( c o t x ) = − 1 s i n 2 x d x = c s c 2 x d x
d(secx)=secx∗tanxdx d ( s e c x ) = s e c x ∗ t a n x d x
( secx=1cosx s e c x = 1 c o s x )
d(cscx)=−cscx∗cotxdx d ( c s c x ) = − c s c x ∗ c o t x d x
( cscx=1sinx c s c x = 1 s i n x )

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d(arcsinx)=11−x2√dx d ( a r c s i n x ) = 1 1 − x 2 d x
d(arccosx)=−11−x2√dx d ( a r c c o s x ) = − 1 1 − x 2 d x
d(arctanx)=11+x2dx d ( a r c t a n x ) = 1 1 + x 2 d x
d(arccotx)=−11+x2dx d ( a r c c o t x ) = − 1 1 + x 2 d x

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