数学 微积分全公式哦(格式全打乱了，哭，但是可以参考一下，没什么用，有点水)(初来论坛，是新生，不知道规矩，有错请佬们多多提醒和包涵)
😘--
### **一、微分学公式**
#### 1. 基本导数规则
- **幂函数**
\frac{d}{dx} x^n = n x^{n-1}
dx
d
x
n
=nx
n−1
- **指数函数**
\frac{d}{dx} e^x = e^x
dx
d
e
x
=e
x
\frac{d}{dx} a^x = a^x \ln a
dx
d
a
x
=a
x
lna
- **对数函数**
\frac{d}{dx} \ln x = \frac{1}{x}
dx
d
lnx=
x
1
\frac{d}{dx} \log_a x = \frac{1}{x \ln a}
dx
d
log
a
x=
xlna
1
- **三角函数**
\frac{d}{dx} \sin x = \cos x
dx
d
sinx=cosx \frac{d}{dx} \cos x = -\sin x
dx
d
cosx=−sinx
\frac{d}{dx} \tan x = \sec^2 x
dx
d
tanx=sec
2
x \frac{d}{dx} \cot x = -\csc^2 x
dx
d
cotx=−csc
2
x
#### 2. 导数运算法则
- **和差法则**
\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)
dx
d
[f(x)±g(x)]=f
′
(x)±g
′
(x)
- **乘积法则**
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
dx
d
[f(x)g(x)]=f
′
(x)g(x)+f(x)g
′
(x)
- **商法则**
\frac{d}{dx} \left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
dx
d
[
g(x)
f(x)
]=
[g(x)]
2
f
′
(x)g(x)−f(x)g
′
(x)
- **链式法则**
\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
dx
d
f(g(x))=f
′
(g(x))⋅g
′
(x)
#### 3. 高阶导数
- 二阶导数：\frac{d^2}{dx^2} f(x) = f''(x)
dx
2
d
2
f(x)=f
′′
(x)
---
### **二、积分学公式**
#### 1. 基本积分公式
- **幂函数**
\int x^n dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)∫x
n
dx=
n+1
x
n+1
+C(n
=−1)
- **指数函数**
\int e^x dx = e^x + C∫e
x
dx=e
x
+C \int a^x dx = \frac{a^x}{\ln a} + C∫a
x
dx=
lna
a
x
+C
- **三角函数**
\int \sin x dx = -\cos x + C∫sinxdx=−cosx+C \int \cos x dx = \sin x + C∫cosxdx=sinx+C
\int \sec^2 x dx = \tan x + C∫sec
2
xdx=tanx+C \int \csc^2 x dx = -\cot x + C∫csc
2
xdx=−cotx+C
#### 2. 积分法则
- **线性性**
\int [af(x) + bg(x)] dx = a\int f(x)dx + b\int g(x)dx∫[af(x)+bg(x)]dx=a∫f(x)dx+b∫g(x)dx
- **分部积分法**
\int u dv = uv - \int v du∫udv=uv−∫vdu
- **换元积分法**
\int f(g(x))g'(x)dx = \int f(u)du∫f(g(x))g
′
(x)dx=∫f(u)du （令 u = g(x)u=g(x)）
#### 3. 定积分与牛顿-莱布尼茨公式
\int_a^b f(x)dx = F(b) - F(a)∫
a
b
f(x)dx=F(b)−F(a) （F(x)F(x) 是 f(x)f(x) 的原函数）
---
### **三、重要定理与公式**
1. **中值定理**
- 罗尔定理：若 f(a)=f(b)f(a)=f(b)，则存在 c \in (a,b)c∈(a,b) 使 f'(c)=0f
′
(c)=0
- 拉格朗日中值定理：存在 c \in (a,b)c∈(a,b) 使 f'(c) = \frac{f(b)-f(a)}{b-a}f
′
(c)=
b−a
f(b)−f(a)
2. **洛必达法则**
当 \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}lim
x→c
g(x)
f(x)
=
0
0
或 \frac{\infty}{\infty}
∞
∞
时：
\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}lim
x→c
g(x)
f(x)
=lim
x→c
g
′
(x)
f
′
(x)
3. **泰勒展开**
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^nf(x)=∑
n=0
∞
n!
f
(n)
(a)
(x−a)
n
4. **常见函数的泰勒级数**
- e^x = \sum_{n=0}^\infty \frac{x^n}{n!}e
x
=∑
n=0
∞
n!
x
n
- \sin x = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}sinx=∑
n=0
∞
(−1)
n
(2n+1)!
x
2n+1
---
### **四、多元微积分公式**
1. **偏导数**
\frac{\partial}{\partial x_i} f(x_1, x_2, ..., x_n)
∂x
i
∂
f(x
1
,x
2
,...,x
n
)
2. **梯度**
\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)∇f=(
∂x
∂f
,
∂y
∂f
,
∂z
∂f
)
3. **二重积分**
\iint_D f(x,y) dxdy∬
D
f(x,y)dxdy
---
### **五、微分方程基础**
1. **分离变量法**
\frac{dy}{dx} = g(x)h(y) \Rightarrow \int \frac{1}{h(y)} dy = \int g(x) dx
dx
dy
=g(x)h(y)⇒∫
h(y)
1
dy=∫g(x)dx
2. **一阶线性微分方程**
解 y' + P(x)y = Q(x)y
′
+P(x)y=Q(x)：
y = e^{-\int P dx} \left( \int Q e^{\int P dx} dx + C \right)y=e
−∫Pdx
(∫Qe
∫Pdx
dx+C)
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