人工智能教程 - 数学基础课程1.1 - 数学分析（一）3-4 导数的乘除运算,链式法则

导数例子

Ex:Specific $f{}'(x) \ \ (f(x)=x^n,\frac{1}{x})$

General:

(u+v)’=u’+v’

(Cu)’ =Cu’ （C为constant）

Specific: 根据 difference quotient

$\frac{d}{dx}sinx = cosx$

$\frac{d}{dx}cosx = -sinx$

General deriv rules- Product rule:

(uv)’ = u’v+uv’

Quotient rule:

$(\frac{u}{v})'=\frac{u'v-uv'}{v^2}$

$\frac{d}{dt} = C \ \frac{du}{dt}$

Composition rule

$y = (sint)^{10} = sin^{10}t$

Chain Rules

${\frac{dy}{dt} = \frac{dy}{dx} . \frac{dx}{dt}}$

EX:

$\frac{d}{dt}(sint)^{10}$

1. x =sint
2. $=10x^9.cos(t)$
3. $= 10(sint(t))^{9}.cos(t)$

Really and truly,once you have the chain rule,the world is yours to conquer.

higher derivative:

u=u(x) u’ u’’ u’’’ $u^{(4)}=(u''')'$

${\color{Red} u''=\frac{d}{dx}.\frac{du}{dx} = \frac{d}{dx} \frac{d}{dx}.u}$

${\color{Red} =(\frac{d}{dx})^2.u}$

${\color{Red} =\frac{d^2}{(dx)^2}.u=\frac{d^2u}{dx^2}}$

EX:

$D^nx^n = n!$

来源：CSDN

作者：KuFun人工智能

链接：https://blog.csdn.net/fsdaewrq/article/details/104348964