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導函數的計算 PowerPoint PPT Presentation


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第四講. 導函數的計算. 課程內容:. 1. 基本函數的導函數 2. 微分法公式 3. 連鎖法則. 課程內容:. 1. 基本函數的導函數 2. 微分法公式 3. 連鎖法則. 因為複雜的代數函數是基本代數函數經由加、減、乘、除、方根或合成運算的結果,因此為了求得複雜函數的導函數,先利用導函數的定義,建立一些基本代數函數的導函數。

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導函數的計算

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ff 'Dx(dee xdee)Dx:ff ' Dxf=f ' Dxf(x)=f '(x)Dx


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4-1

(a) f(x)=kkf'(x)=0Dx(k)=0

(b)f(x)=xf'(x)=1Dx(x)=1

(c)f(x)=xnnf(x)=nxn-1Dx(xn)=nxn-1

(a)(b)(c)

(power rule)Dx(xn)=nxn-1


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(a)(b)(c)

(power rule)Dx(xn)=nxn-1


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f '(c)f(x)x=cf(x)f(x)


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1f(x)=x2(-)

4-1(c)f '(x)=2x(-)f(x)(-)


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2 (0)

(0)f(x)(0)


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3 (-0)(0)

(-0)(0)f(x)(-0)(0)


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4f(x)=|x|

f '(x)f '(x)f(x)

f '(x)(-0)(0)

f(x)(-0)(0)


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(a)y=|x|(0,0)f'(0)x>0y=|x|y=x1f'(x)=1x<0y=|x|y=-x-1f'(x)=-1 (b)


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4-2

f'(c)fcfc

fc


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fcc4f(x)=|x|f '(0)f0


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5f(x)=[x]

c

f '(c)f(x)=[x]


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6 0

ff'

f'(0) 0


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(i) 4f(x)=|x|x=0(ii) 5f(x)=[x] (iii) 6 y


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4-3

fgk

(a) kf(kf)'(x)=kf'(x)Dx[kf(x)]=kDxf(x)

(b) f+g(f+g)'(x)=f'(x)+g'(x)Dx[f(x)+g(x)]=Dxf(x)+Dxg(x)

(c) f-g(f-g)'(x)=f'(x)-g'(x)Dx[f(x)-g(x)]=Dxf(x)-Dxg(x)

(d) fg(fg)'(x)=f(x)g'(x)+ f'(x)g(x)Dx[f(x)g(x)]= f(x)Dxg(x)+ (Dxf(x))g(x)(product rule)

(e)

g(x)0(quotient rule)


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(a)(b)(c)(d)(e)

(d)fg


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(e)fg


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4-14-3


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7f(x)=x5+2x4-4x3+x2+5x+1f'(x)

4-14-3(a)(b)(c)f'(x)

f'(x)=Dxf(x)=Dx(x5+2x4-4x3+x2+5x+1)

=Dx(x5)+2Dx(x4)-4Dx(x3)+Dx(x2)+5Dx(x)+Dx(1)

=5x4+8x3-12x2+2x+5


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8 f'(x)

4-3(a)(b)(e)4-1

f'(x)


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9f(x)=(2x12)(5x8)f'(x)

4-3(a)(d)4-1 f'(x)

f'(x)=Dx[(2x12)(5x8)]=(2x12)Dx(5x8)+[Dx(2x12)](5x8)

=(2x12)(40x7)+(24x11)(5x8)

=80x19+120x19=200x19

f'(x)=Dx[(2x12)(5x8)]=Dx[10x20]=200x19


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10f(x)=(x5+x4+x3+x2+x+1)(2x2-x-5)f'(x)

f'(x)=Dx[(x5+x4+x3+x2+x+1)(2x2-x-5)]

= (x5+x4+x3+x2+x+1)Dx(2x2-x-5)

+[Dx(x5+x4+x3+x2+x+1)](2x2-x-5)

= (x5+x4+x3+x2+x+1)(4x-1)

+ (5x4+4x3+3x2+2x+1)(2x2-x-5)


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910(transcendental function)

nDx(xn)=nxn-1

n


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4-4

nDx(xn)=nxn-1

n04-1nxn


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11 y'

y'


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2.

3.


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(chain rule)


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h(x)=(x2+x+1)20h(x)hg(x)=x20f(x)=x2+x+1h(x)=g(f(x))


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y=f(x)primef'f'(x)y'deeDxfDxf(x)Dxy


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xx1x2xx2-x1(increment)x(delta x)x=x2-x1xx


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y=f(x)xx+hx=(x+h)-x=hf(x)f(x+h)y=f(x+h)-f(x)

(x,f(x))(x+x,f(x+x))x0


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dee y dee xdydx Dx


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12y=x5-3x3+5x-10

4-14-3


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13


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fgxfugyy=g(u)u=f(x)DxufDuygDuyDxuDxyDxy=Duy Dxu


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4-5

gfy=(g f )(x)fxgu=f(x)g fx (g f )'(x)=

g'(f(x))f'(x)Dxy=DuyDxu


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g'(u) ug(u+u)-g(u)=g'(u) u+u (u)u=f(x)u=f(x+x)-f(x)g(f(x+x))-g(f(x))=g'(f(x))(f(x+x)-f(x))

+(f(x+x)-f(x)) (x)x


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(g f )'(x)=g'(f(x))f'(x)deeDxy=DuyDxu


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4-5 prime

deeduy=h(u)u=g(w)w=f(x)


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14h(x)=(x2+x+1)20h'(x)

hy=g(u)=u20u=f(x)=x2+x+1y=g(f(x))=h(x)

ux


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15r(x)=[(x2+2)10+5(x2+2)]6r'(x)

ry=h(u)=u6u=g(w)=

w10+5ww=f(x)=x2+2y=h(g(f(x)))=

r(x)

uwx


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(exponent)


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4-6

rDx(xr)=rxr-1xxr

r pqp


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16 f'(x)

f

4-6


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