导数与微分-ZhiMap思维导图

导数与微分
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- 定义
  - 导数
    - $\Delta y = f(x + \Delta x) - f(x) \\ \lim\limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}存在$ ，记为 $f'(x_0)或\dfrac{dy }{dx}|_{x=x_0}$
  - 可微
    - $\Delta y=f\left(x_{0}+\Delta x\right)-f\left(x_{0}\right)\left(=f(x)-f\left(x_{0}\right), \Delta x=x-x_{0}\right)$ ，记为 $dy |_{x = x_0} = A \Delta x 或dy |_{x = x_0} = A dx$
    - 条件
      - $\Delta y=A \Delta x+o( \Delta x)$
- Notes
  - 导数
    - $\displaystyle f^{\prime}(a)=\lim _{\Delta x \rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}$
    - $f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$
    - $\Delta x \rightarrow 0\left\{\begin{array}{l} \Delta x \rightarrow 0^{-} \\ \Delta x \rightarrow 0^{+} \end{array}\right.$
    - $x \rightarrow a\left\{\begin{array}{l} x \rightarrow a^{+} \\ x \rightarrow a^{-} \end{array}\right.$
    - 左导数： $\lim _{\Delta x \rightarrow 0^{-}} \frac{\Delta y}{\Delta x}=\lim _{x \rightarrow a^{-}} \frac{f(x)-f(a)}{x-a}\overset{def}{==}f'_-(a)$ $\displaystyle \lim _{\Delta x \rightarrow 0^{-}} \frac{\Delta y}{\Delta x}=\lim _{x \rightarrow a^{-}} \frac{f(x)-f(a)}{x-a} \stackrel{\text {def}}{=}=f_{-}^{\prime}(a)$
    - 右导数： $\displaystyle \lim _{\Delta x \rightarrow 0^{+}} \frac{\Delta y}{\Delta x}=\lim _{x \rightarrow a^{+}} \frac{f(x)-f(a)}{x-a} \stackrel{\text {def}}{=}=f_{+}^{\prime}(a)$
    - $f'(a)\exists \Leftrightarrow f'_-(a)\wedge f'_+(a)\wedge f'_-(a) = f'_+(a)$
    - $f(x)在x= a点可导\Rightarrow f(x)在x=a点处连续\\ f(x)在x=a点处连续 \ne f(x)在x= a点可导$
    - $\displaystyle f(x)连续\wedge \lim _{x \rightarrow a} \frac{f(x)-b}{x-a} \Rightarrow \left\{\begin{array}{l} f(a)=b \\ f^{\prime}(a) = A \end{array}\right.$
  - 微分
    - $f(x)在x = x_0可到\Leftrightarrow f(x)在x = x_0处可微$
    - $f(x)在x = x_0处可微 \Leftrightarrow f(x)在x= x_0处连续$
    - $y = f(x)处处可导\Rightarrow dy = df(x) = f'(x) dx$
- 求导工具
  - 基本公式
    - $(c)' = 0$
    - $(x^a)' = a x^{a - 1}$
    - $(a^x)' = a ^x \ln a([e^x]' = e^x)$
    - $\log _{a} x=\frac{1}{x\ln a} ([\ln x]'=\frac{1}{x})$
    - 三角函数
      - $\begin{aligned} &(\sin x)'=\cos x\\ &(\cos x)^{\prime}=-\sin x\\ &(\tan x)^{\prime}=\sec ^{2} x\\ &(\cot x)^{\prime}=-\csc^{2} x\\ &(\sec x)^{\prime}=\sec x \tan x\\ &(\cot x )^{\prime}=-\csc x \cot x \end{aligned}$
    - 反三角函数
      - $\begin{array}{l} (\arcsin x)'=\frac{1}{\sqrt{1-x^{2}}} \\ (\arccos x)'=\frac{1}{-\sqrt{1-x^{2}}} \\ (\arctan x)=\frac{1}{1+x^{2}} \\ ( \cot x)^{\prime}=-\frac{1}{1+x^2} \end{array}$
  - 求导方法
    - 四则运算求导
      - $\begin{array}{l} (u \pm v)' = u' \pm v'\\ (uv)' = u'v + uv'\\ (ku)' = ku'\\ (uvw)' = u'vw + uv'w+ uvw'\\ \dfrac{u}{v} = \dfrac{u'v - uv'}{v^2} \end{array}$
    - 链式求导法则
      - $\displaystyle \frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}=f^{\prime}(u) \cdot \varphi^{\prime}(x)=f^{\prime}[\varphi(x)] \varphi^{\prime}(x)$
    - 反函数求导
      - $[f^{-1}(x)]' = \dfrac{1}{f'(x)}$
      - 证明原理： $\Delta y = O(\Delta x)$
    - 隐函数求导
    - 参数方程求导
      - $\displaystyle \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi'(t)}{\varphi'(t)}$ ``
- 重要导数
  - $(u \pm v) ^{(n)} = u^{(n)} \pm v^{(n)}$
  - $(uv) ^{(n)} = C_n^0 u^{(n)}v + C_n^1 u^{(n - 1) }v' + \cdots + C_n^n uv^{(n)}$
  - $(\sin x)^{(n)} = sin (x + \dfrac{n\pi}{2})$
  - $(\cos x)^{(n)} = \cos (x + \dfrac{n\pi}{2})$
  - $\displaystyle (\frac{1}{ax+ b})^{(n)} = \frac{(-1)^n n! a^n}{(ax+b)^{n + 1}}$

导数与微分

​定义

​导数

​ Δy=f(x+Δx)−f(x)lim⁡Δx→0ΔyΔx存在\Delta y = f(x + \Delta x) - f(x) \\ \lim\limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}存在Δy=f(x+Δx)−f(x)Δx→0lim​ΔxΔy​存在 ，记为 f′(x0)或dydx∣x=x0f'(x_0)或\dfrac{dy }{dx}|_{x=x_0}f′(x0​)或dxdy​∣x=x0​​

​可微

​条件

​ Δy=AΔx+o(Δx)\Delta y=A \Delta x+o( \Delta x)Δy=AΔx+o(Δx)

​Notes

​导数

​ f′(a)=lim⁡Δx→0f(a+Δx)−f(a)Δx\displaystyle f^{\prime}(a)=\lim _{\Delta x \rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}f′(a)=Δx→0lim​Δxf(a+Δx)−f(a)​

​ f′(a)=lim⁡x→af(x)−f(a)x−af^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}f′(a)=limx→a​x−af(x)−f(a)​

​ Δx→0{Δx→0−Δx→0+\Delta x \rightarrow 0\left\{\begin{array}{l} \Delta x \rightarrow 0^{-} \\ \Delta x \rightarrow 0^{+} \end{array}\right.Δx→0{Δx→0−Δx→0+​

​ x→a{x→a+x→a−x \rightarrow a\left\{\begin{array}{l} x \rightarrow a^{+} \\ x \rightarrow a^{-} \end{array}\right.x→a{x→a+x→a−​

​ f′(a)∃⇔f−′(a)∧f+′(a)∧f−′(a)=f+′(a)f'(a)\exists \Leftrightarrow f'_-(a)\wedge f'_+(a)\wedge f'_-(a) = f'_+(a)f′(a)∃⇔f−′​(a)∧f+′​(a)∧f−′​(a)=f+′​(a)

​ f(x)在x=a点可导⇒f(x)在x=a点处连续f(x)在x=a点处连续̸=f(x)在x=a点可导f(x)在x= a点可导\Rightarrow f(x)在x=a点处连续\\ f(x)在x=a点处连续 \ne f(x)在x= a点可导f(x)在x=a点可导⇒f(x)在x=a点处连续f(x)在x=a点处连续̸=f(x)在x=a点可导

​ f(x)连续∧lim⁡x→af(x)−bx−a⇒{f(a)=bf′(a)=A\displaystyle f(x)连续\wedge \lim _{x \rightarrow a} \frac{f(x)-b}{x-a} \Rightarrow \left\{\begin{array}{l} f(a)=b \\ f^{\prime}(a) = A \end{array}\right.f(x)连续∧x→alim​x−af(x)−b​⇒{f(a)=bf′(a)=A​

​微分

​ f(x)在x=x0可到⇔f(x)在x=x0处可微f(x)在x = x_0可到\Leftrightarrow f(x)在x = x_0处可微f(x)在x=x0​可到⇔f(x)在x=x0​处可微

​ f(x)在x=x0处可微⇔f(x)在x=x0处连续f(x)在x = x_0处可微 \Leftrightarrow f(x)在x= x_0处连续f(x)在x=x0​处可微⇔f(x)在x=x0​处连续

y=f(x)处处可导⇒dy=df(x)=f′(x)dxy = f(x)处处可导\Rightarrow dy = df(x) = f'(x) dxy=f(x)处处可导⇒dy=df(x)=f′(x)dx

​求导工具

​基本公式

​ (c)′=0(c)' = 0 (c)′=0

​ (xa)′=axa−1(x^a)' = a x^{a - 1} (xa)′=axa−1

​ (ax)′=axln⁡a([ex]′=ex)(a^x)' = a ^x \ln a([e^x]' = e^x) (ax)′=axlna([ex]′=ex)

​ log⁡ax=1xln⁡a([ln⁡x]′=1x)\log _{a} x=\frac{1}{x\ln a} ([\ln x]'=\frac{1}{x})loga​x=xlna1​([lnx]′=x1​)

​三角函数

​反三角函数

求导方法

​四则运算求导

​链式求导法则

​ dydx=dydu⋅dudx=f′(u)⋅φ′(x)=f′[φ(x)]φ′(x)\displaystyle \frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}=f^{\prime}(u) \cdot \varphi^{\prime}(x)=f^{\prime}[\varphi(x)] \varphi^{\prime}(x)dxdy​=dudy​⋅dxdu​=f′(u)⋅φ′(x)=f′[φ(x)]φ′(x)

​反函数求导

​ [f−1(x)]′=1f′(x)[f^{-1}(x)]' = \dfrac{1}{f'(x)}[f−1(x)]′=f′(x)1​

​证明原理： Δy=O(Δx)\Delta y = O(\Delta x)Δy=O(Δx)

​隐函数求导

​参数方程求导

​ dydx=dy/dtdx/dt=ψ′(t)φ′(t)\displaystyle \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi'(t)}{\varphi'(t)}dxdy​=dx/dtdy/dt​=φ′(t)ψ′(t)​ ``

​重要导数

​ (u±v)(n)=u(n)±v(n)(u \pm v) ^{(n)} = u^{(n)} \pm v^{(n)}(u±v)(n)=u(n)±v(n)

​ (uv)(n)=Cn0u(n)v+Cn1u(n−1)v′+⋯+Cnnuv(n)(uv) ^{(n)} = C_n^0 u^{(n)}v + C_n^1 u^{(n - 1) }v' + \cdots + C_n^n uv^{(n)}(uv)(n)=Cn0​u(n)v+Cn1​u(n−1)v′+⋯+Cnn​uv(n)

​ (sin⁡x)(n)=sin(x+nπ2)(\sin x)^{(n)} = sin (x + \dfrac{n\pi}{2})(sinx)(n)=sin(x+2nπ​)

​ (cos⁡x)(n)=cos⁡(x+nπ2)(\cos x)^{(n)} = \cos (x + \dfrac{n\pi}{2})(cosx)(n)=cos(x+2nπ​)

​ (1ax+b)(n)=(−1)nn!an(ax+b)n+1\displaystyle (\frac{1}{ax+ b})^{(n)} = \frac{(-1)^n n! a^n}{(ax+b)^{n + 1}}(ax+b1​)(n)=(ax+b)n+1(−1)nn!an​

定义

导数

$\Delta y = f(x + \Delta x) - f(x) \\ \lim\limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}存在$ ，记为 $f'(x_0)或\dfrac{dy }{dx}|_{x=x_0}$

可微

条件

$\Delta y=A \Delta x+o( \Delta x)$

Notes

导数

$\displaystyle f^{\prime}(a)=\lim _{\Delta x \rightarrow 0} \frac{f(a+\Delta x)-f(a)}{\Delta x}$

$f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a}$

$\Delta x \rightarrow 0\left\{\begin{array}{l} \Delta x \rightarrow 0^{-} \\ \Delta x \rightarrow 0^{+} \end{array}\right.$

$x \rightarrow a\left\{\begin{array}{l} x \rightarrow a^{+} \\ x \rightarrow a^{-} \end{array}\right.$

$f'(a)\exists \Leftrightarrow f'_-(a)\wedge f'_+(a)\wedge f'_-(a) = f'_+(a)$

$f(x)在x= a点可导\Rightarrow f(x)在x=a点处连续\\ f(x)在x=a点处连续 \ne f(x)在x= a点可导$

$\displaystyle f(x)连续\wedge \lim _{x \rightarrow a} \frac{f(x)-b}{x-a} \Rightarrow \left\{\begin{array}{l} f(a)=b \\ f^{\prime}(a) = A \end{array}\right.$

微分

$f(x)在x = x_0可到\Leftrightarrow f(x)在x = x_0处可微$

$f(x)在x = x_0处可微 \Leftrightarrow f(x)在x= x_0处连续$

$y = f(x)处处可导\Rightarrow dy = df(x) = f'(x) dx$

求导工具

基本公式

$(c)' = 0$

$(x^a)' = a x^{a - 1}$

$(a^x)' = a ^x \ln a([e^x]' = e^x)$

$\log _{a} x=\frac{1}{x\ln a} ([\ln x]'=\frac{1}{x})$

三角函数

反三角函数

四则运算求导

链式求导法则

$\displaystyle \frac{d y}{d x}=\frac{d y}{d u} \cdot \frac{d u}{d x}=f^{\prime}(u) \cdot \varphi^{\prime}(x)=f^{\prime}[\varphi(x)] \varphi^{\prime}(x)$

反函数求导

$[f^{-1}(x)]' = \dfrac{1}{f'(x)}$

证明原理： $\Delta y = O(\Delta x)$

隐函数求导

参数方程求导

$\displaystyle \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{\psi'(t)}{\varphi'(t)}$ ``

重要导数

$(u \pm v) ^{(n)} = u^{(n)} \pm v^{(n)}$

$(uv) ^{(n)} = C_n^0 u^{(n)}v + C_n^1 u^{(n - 1) }v' + \cdots + C_n^n uv^{(n)}$

$(\sin x)^{(n)} = sin (x + \dfrac{n\pi}{2})$

$(\cos x)^{(n)} = \cos (x + \dfrac{n\pi}{2})$

$\displaystyle (\frac{1}{ax+ b})^{(n)} = \frac{(-1)^n n! a^n}{(ax+b)^{n + 1}}$