数学基础知识总结 |
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文章目录
基本导数公式基本导数运算法则加法运算减法运算乘法运算除法运算带有常数C的导数
微分的四则运算加减法计算带有常数的微分乘法计算除法计算
基本导数公式
原函数
f
(
x
)
f(x)
f(x)导数
f
′
(
x
)
f'(x)
f′(x)C (C为常数)0
x
n
x^n
xn
n
x
n
−
1
nx^{n-1}
nxn−1
C
x
C^x
Cx
C
x
l
n
C
C^xlnC
CxlnC (C为常数,且大于0)
e
x
e^x
ex (e为自然常数)
e
x
(
l
n
e
)
=
e
x
⋅
1
=
e
x
e^x(lne) = e^x \cdot 1 = e^x
ex(lne)=ex⋅1=ex
l
o
g
c
x
log_cx
logcx
l
o
g
a
e
x
\frac{log_ae}{x}
xlogae
l
n
x
ln x
lnx
1
x
\frac{1}{x}
x1
s
i
n
x
sin x
sinx
c
o
s
x
cos x
cosx
c
o
s
x
cos x
cosx
−
s
i
n
x
-sin x
−sinx
t
a
n
x
tan x
tanx
s
e
c
2
x
=
1
c
o
s
2
x
sec^2x = \frac{1}{cos^2 x}
sec2x=cos2x1
c
o
t
x
cot x
cotx
−
c
s
c
2
x
=
−
1
s
i
n
2
x
-csc^2 x = -\frac{1}{sin^2 x}
−csc2x=−sin2x1
基本导数运算法则
加法运算
F ′ ( x ) = ( f ( x ) + g ( x ) ) ′ = f ′ ( x ) + g ′ ( x ) F'(x)=(f(x) + g(x))' = f'(x) + g'(x) F′(x)=(f(x)+g(x))′=f′(x)+g′(x) 减法运算F ′ ( x ) = ( f ( x ) − g ( x ) ) ′ = f ′ ( x ) − g ′ ( x ) F'(x)=(f(x) - g(x))' = f'(x) - g'(x) F′(x)=(f(x)−g(x))′=f′(x)−g′(x) 乘法运算F ′ ( x ) = ( f ( x ) × g ( x ) ) ′ = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) F'(x)=(f(x) \times g(x))' = f'(x)g(x) + f(x)g'(x) F′(x)=(f(x)×g(x))′=f′(x)g(x)+f(x)g′(x) 除法运算F ′ ( x ) = { f ( x ) g ( x ) } ′ = { f ( x ) ′ g ( x ) − f ( x ) g ( x ) ′ g 2 ( x ) } F'(x) = \left \{ \frac{f(x)}{g(x)} \right \}' = \left \{ \frac{f(x)'g(x) - f(x)g(x)'}{g^2(x)} \right \} F′(x)={g(x)f(x)}′={g2(x)f(x)′g(x)−f(x)g(x)′} 带有常数C的导数F ′ ( x ) = ( C ⋅ f ( x ) ) ′ = C ⋅ f ( x ) ′ F'(x) = (C \cdot f(x))' = C \cdot f(x)' F′(x)=(C⋅f(x))′=C⋅f(x)′ 微分的四则运算微分常见的表示符号有三种,在偏微分方程中,以 ∂ \partial ∂表示,在通常则是以 d d d表示,某些教科书上也有以 d i f f ( x ) diff(x) diff(x)进行表示,代表一种计算方法, d x dx dx表达的含义与通常 f ( x ) f(x) f(x)是一样的,因为数学家比较懒的原因, d ( x ) d(x) d(x)就约定俗成的用 d x dx dx进行表达了。 加减法计算d ( f ( x ) ± g ( x ) ) = d ( f ( x ) ) ± d ( g ( x ) ) d(f(x) \pm g(x)) = d(f(x)) \pm d(g(x)) d(f(x)±g(x))=d(f(x))±d(g(x)) 带有常数的微分d ( C f ( x ) ) = C ⋅ d ( f ( x ) ) d(Cf(x)) = C \cdot d(f(x)) d(Cf(x))=C⋅d(f(x)) 乘法计算d ( f ( x ) g ( x ) ) = d ( f ( x ) ) g ( x ) + f ( x ) d ( g ( x ) ) d(f(x)g(x)) = d(f(x))g(x) + f(x)d(g(x)) d(f(x)g(x))=d(f(x))g(x)+f(x)d(g(x)) 除法计算d { f ( x ) g ( x ) } = { d f ( x ) g ( x ) − f ( x ) d g ( x ) g 2 ( x ) } d \left \{ \frac{f(x)}{g(x)} \right \} = \left \{ \frac{df(x)g(x) - f(x)dg(x)}{g^2(x)} \right \} d{g(x)f(x)}={g2(x)df(x)g(x)−f(x)dg(x)} |
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