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ͧ١
ԪŤ ١ (ѧ: Chain rule) ٵѺ͹ؾѹͧѧѹԵ

Ѵ ҡ y ¹ŧ u ¹ŧ x ѵҡ¹ŧͧ y ºѺ x ҡŤٳ ͧѵҡ¹ŧͧ y ºѺ u ٳѺ ѵҡ¹ŧͧ u ºѺ x

餹˹觻չҴѵ 0.5 õͪ سŴŧдѺ٧ ѵ Ŵŧ 6 °F ͡ Ҥٳ 6 °F ͡ô 0.5 õͪ 3 °F ͪ äӹdz蹹繵ҧͧûء顮١

㹷ҧժԵ ١ (Ѻ) к ҿѧѹ f ͹ؾѹ g(x) пѧѹ g ͹ؾѹ x Ҩ f \circ g = f(g(x)) ѧ


 \frac {df} {dx} = \frac {d} {dx} f(g(x)) = f

͡ҡ ѭóͧźԫ ١¹᷹ѧ:


\frac {df}{dx} = \frac {df} {dg} \frac {dg}{dx}

\frac {df} {dg} к f ¹ŧ g ͹繵˹.

Ծѹ ǹѺͧ١һԾѹ¡᷹

[͹]

[] The general power rule

Ţ¡ѧöѺ١

[] Example I

Ԩó f(x) = (x2 + 1)3. f(x) ºѺ h(g(x)) · g(x) = x2 + 1 h(x) = x3 ѧ

f'(x) = 3(x2 + 1)2(2x)
= 6x(x2 + 1)2

[] Example II

㹡͹ؾѹͧѧѹ⡳Ե

f(x) = sin(x2),

ö¹ f(x) = h(g(x)) h(x) = sinx g(x) = x2 ҡ١

f'(x) = 2xcos(x2)

ͧҡ h'(g(x)) = cos(x2) g'(x) = 2x

[] ١Ѻµ

١Ѻѧѹµ蹡ѹ ҧ տѧѹ f(u(x,y),v(x,y)) ·

u(x,y) = 3x + y2 v(x,y) = sin(xy)

ѧ

 {\partial f \over \partial x}={\partial f \over \partial u}{\partial u \over \partial x}+{\partial f \over \partial v}{\partial v \over \partial x}=3 + \cos(xy)y

[] ٨졮١

f g 繿ѧѹ x 繨ӹǹ f ö͹ؾѹ g(x) g ͹ؾѹ x ѧ ҡͧ͹ؾѹ

 g(x+\delta)-g(x)= \delta g  \frac{\epsilon(\delta)}{\delta} \to 0 \, з \delta\to 0.

㹷ӹͧǡѹ

 f(g(x)+\alpha) - f(g(x)) = \alpha f \frac{\eta(\alpha)}{\alpha} \to 0 \, з \alpha\to 0. \,

 f(g(x+\delta))-f(g(x))\, = f(g(x) + \delta g
 = \alpha_\delta f

\alpha_\delta = \delta g Ңз \delta\to 0 \frac{\alpha_\delta}{\delta}\to g \frac{\eta(\alpha_\delta)}{\delta}\to 0 ѧ

 \frac{f(g(x+\delta))-f(g(x))}{\delta} \to g з \delta \to 0

[] ١鹰ҹ

١繤سѵԾ鹰ҹͧͧ͹ؾѹ E F G ԺҹҤ (件֧٤Դ) f: EF g: FG 繿ѧѹ ж x Ҫԡͧ E f ͹ؾѹ x g ͹ؾѹ f(x) ͹ؾѹ (͹ؾѹવ) ͧѧѹԵ g o f x 繴ѧ

\mbox{D}_x\left(g \circ f\right) = \mbox{D}_{f\left(x\right)}\left(g\right) \circ \mbox{D}_x\left(f\right).

ѧࡵ͹ؾѹŧԧ Ţ ҡŧԧ᷹ԡ (¹ԡ) ҧҹҨС繡äٳԡ

á˹١ѴਹöҡԸշ繷ҡش M N P Ŵ Ck (ͺҹҤŴ)

f: MN g: NP

繡ŧ͹ؾѹ ͹ؾѹͧ f ᷹ df 繡ŧҡʢͧ M ѧʢͧ N ö¹᷹

\mbox{d}\left(g \circ f\right) = \mbox{d}g \circ \mbox{d}f.

Ըչ ٻẺͧ͹ؾѹлʨж١ͧٻѧ Category ͧŴ C աŧ C ѳҹ

[] ෹Ѻ١

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