人工智能教程 - 數學基礎課程1.1 - 數學分析(一)3-4 導數的乘除運算,鏈式法則

導數例子

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

(u+v)’=u’+v’

(Cu)’ =Cu’ (C爲constant)

Specific: 根據 difference quotient

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

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

General deriv rules- Product rule:

(uv)’ = u’v+uv’

Quotient rule:

(uv)=uvuvv2(\frac{u}{v})'=\frac{u'v-uv'}{v^2}

ddt=C dudt\frac{d}{dt} = C \ \frac{du}{dt}

Composition rule

y=(sint)10=sin10ty = (sint)^{10} = sin^{10}t

Chain Rules

dydt=dydx.dxdt{\frac{dy}{dt} = \frac{dy}{dx} . \frac{dx}{dt}}

EX:

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

  1. x =sint
  2. =10x9.cos(t)=10x^9.cos(t)
  3. =10(sint(t))9.cos(t)= 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)u^{(4)}=(u''')'

u=ddx.dudx=ddxddx.u{\color{Red} u''=\frac{d}{dx}.\frac{du}{dx} = \frac{d}{dx} \frac{d}{dx}.u}

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

=d2(dx)2.u=d2udx2{\color{Red} =\frac{d^2}{(dx)^2}.u=\frac{d^2u}{dx^2}}

EX:

Dnxn=n!D^nx^n = n!

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