人工智能教程 - 数学基础课程1.1 - 数学分析（一）1-2 导数,二项式定理

几何观点下的导数:

$y-y_{0} = -\frac{1}{x_{0}^{2}}(x-x_{0})$

表示法：

${f}' = \frac{df}{dx} = \frac{dy}{dx}= \frac{d}{dx} f= \frac{d}{dx} y$

f’–newton 牛顿表示法

others - leibuniz 莱伯尼兹表示法

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binomial theorem 二项式定理

$(x+\Delta x)^{n} =(x+\Delta x)..(x+\Delta x)=x^{n}+nx^{n-1}\Delta x+junk(O(\Delta x)^{2})$

$\frac{\Delta f}{\Delta x} = \frac{1}{\Delta x}((x+\Delta x)^2-x^n)$

…
$=nx^{^{n-1}}+O(\Delta x)$

So:

$\frac{d}{dx}x^n = nx^{n-1}$

extends to polynomials:

$\frac{d}{dx}(x^3+5x^{10})=3x^2+50x^9$

什么是导数(Derivative)?

答案:改变率(Rate of change )

average change 瞬时率(instaneous rate)

Ex:

1.q = charge 电荷,$\frac{dq}{dt} =current$ 电流

2.s = distance ,$\frac{ds}{dt}=speed$

Ex:
$h = 80-5t^2$

t=0,h=80
t=4,h=0
ava speed :

$\frac{\Delta h}{\Delta t}=\frac{0-80}{4-0}=-20 m/sec$

instaneous speed:

$\frac{d}{dt}h = 0-10t$

t=4
${h}' =-40m/sec$ 是两倍的ava speed

3.T = temperature

$\frac{dT}{dx}=gradient$

4.测试灵敏度 sensitivity of measures

limits and continuity

• easy limit

• Derivatives are always harder

$\lim_{x\rightarrow x_{0}{}}\frac{f(x_0+\Delta x)-f(x_0)}{x-x_0}; \ \ \ x=x_0,gives \frac{0}{0}$

need concellation
judgement
$1.\lim_{x\rightarrow x_{0}{}}f(x) \ \ \ exists$
$2.f(x_0) \ is \ defined.$
3.they are equal.

$\lim_{x\rightarrow 0{}} \frac{sinx}{x} =1$

$\lim_{x\rightarrow 0{}} \frac{1-cosx}{x} =0$

removable discontinuity:$y=\frac{1}{x}$
点不等
infinite discontinuity:$y=sin\frac{1}{x}$

THEOREM:

DIFF----->CTS 可导必连续

来源：CSDN

作者：KuFun人工智能

链接：https://blog.csdn.net/fsdaewrq/article/details/104348509