á¤Å¤ÙÅÑÊ http://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%A5%E0%B8%B9%E0%B8%81%E0%B9%82%E0%B8%8B%E0%B9%88

¡¯¢Í§ÅÙ¡â«è
ã¹ÇÔªÒá¤Å¤ÙÅÑÊ ¡¯ÅÙ¡â«è (Íѧ¡ÄÉ: Chain rule) ¤×ÍÊÙµÃÊÓËÃѺ¡ÒÃËÒ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹¤ÍÁâ¾ÊÔµ

àËç¹ä´éªÑ´ÇèÒ ËÒ¡µÑÇá»Ã y à»ÅÕè¹á»Å§µÒÁµÑÇá»Ã u «Öè§à»ÅÕè¹á»Å§µÒÁµÑÇá»Ã x áÅéÇ ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§¢Í§ y à·Õº¡Ñº x ËÒä´é¨Ò¡¼Å¤Ù³ ¢Í§ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§¢Í§ y à·Õº¡Ñº u ¤Ù³¡Ñº ÍѵÃÒ¡ÒÃà»ÅÕè¹á»Å§¢Í§ u à·Õº¡Ñº x

ÊÁÁµÔãË餹˹Öè§»Õ¹à¢Ò´éÇÂÍѵÃÒ 0.5 ¡ÔâÅàÁµÃµèͪÑèÇâÁ§ ÍØ³ËÀÙÁÔ¨ÐÅ´µèÓŧàÁ×èÍÃдѺ¤ÇÒÁÊÙ§à¾ÔèÁ¢Öé¹ ÊÁÁµÔãËéÍѵÃÒà»ç¹ Ŵŧ 6 °F µèÍ¡ÔâÅàÁµÃ ¶éÒàÃÒ¤Ù³ 6 °F µèÍ¡ÔâÅàÁµÃ´éÇ 0.5 ¡ÔâÅàÁµÃµèͪÑèÇâÁ§ ¨Ðä´é 3 °F µèͪÑèÇâÁ§ ¡Òäӹdzàªè¹¹Õéà»ç¹µÑÇÍÂèÒ§¢Í§¡ÒûÃÐÂØ¡µìãªé¡®ÅÙ¡â«è

ã¹·Ò§¾Õª¤³Ôµ ¡®ÅÙ¡â«è (ÊÓËÃѺµÑÇá»Ãà´ÕÂÇ) ÃкØÇèÒ ¶éҿѧ¡ìªÑ¹ f ËÒ͹ؾѹ¸ìä´é·Õè g(x) áÅпѧ¡ìªÑ¹ g ËÒ͹ؾѹ¸ìä´é·Õè x ¤×ÍàÃÒ¨Ðä´é f \circ g = f(g(x)) ´Ñ§¹Ñé¹


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

¹Í¡¨Ò¡¹Õé ´éÇÂÊÑ­¡Ã³ì¢Í§äźì¹Ô« ¡®ÅÙ¡â«èà¢Õ¹᷹ä´é´Ñ§¹Õé:


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

àÁ×èÍ \frac {df} {dg} ÃкØÇèÒ f à»ÅÕè¹á»Å§µÒÁ g àËÁ×͹à»ç¹µÑÇá»Ã˹Öè§.

㹡ÒÃËÒ»ÃԾѹ¸ì Êèǹ¡ÅѺ¢Í§¡®ÅÙ¡â«è¤×Í¡ÒÃËÒ»ÃԾѹ¸ìâ´Â¡ÒÃá·¹¤èÒ

à¹×éÍËÒ

[«è͹]

[á¡é] The general power rule

¡®àŢ¡¡ÓÅѧ·ÑèÇä»ÊÒÁÒö¹ÓÁÒãªé¡Ñº¡®ÅÙ¡â«èä´é

[á¡é] Example I

¾Ô¨ÒÃ³Ò f(x) = (x2 + 1)3. f(x) à·Õºä´é¡Ñº h(g(x)) â´Â·Õè g(x) = x2 + 1 áÅÐ h(x) = x3 ´Ñ§¹Ñé¹

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

[á¡é] Example II

㹡ÒÃËÒ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹µÃÕ⡳ÁÔµÔ

f(x) = sin(x2),

àÃÒÊÒÁÒöà¢Õ¹ f(x) = h(g(x)) ´éÇ h(x) = sinx áÅÐ g(x) = x2 ¨Ò¡¡®ÅÙ¡â«è ¨Ðä´é

f'(x) = 2xcos(x2)

à¹×èͧ¨Ò¡ h'(g(x)) = cos(x2) áÅÐ g'(x) = 2x

[á¡é] ¡®ÅÙ¡â«èÊÓËÃѺËÅÒµÑÇá»Ã

¡®ÅÙ¡â«èãªéä´é¡Ñº¿Ñ§¡ìªÑ¹ËÅÒµÑÇá»Ãàªè¹¡Ñ¹ µÑÇÍÂèÒ§àªè¹ ¶éÒàÃÒÁտѧ¡ìªÑ¹ f(u(x,y),v(x,y)) â´Â·Õè

u(x,y) = 3x + y2 áÅÐ v(x,y) = sin(xy)

´Ñ§¹Ñé¹

 {\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

[á¡é] º·¾ÔÊÙ¨¹ì¡®ÅÙ¡â«è

ãËé f áÅÐ g à»ç¹¿Ñ§¡ìªÑ¹ áÅÐãËé x à»ç¹¨Ó¹Ç¹·Õè f ÊÒÁÒöËÒ͹ؾѹ¸ìä´é·Õè g(x) áÅÐ g ËÒ͹ؾѹ¸ìä´é·Õè x ´Ñ§¹Ñé¹ ¨Ò¡¹ÔÂÒÁ¢Í§¡ÒÃËÒ͹ؾѹ¸ìä´é ¨Ðä´é

 g(x+\delta)-g(x)= \delta g «Öè§  \frac{\epsilon(\delta)}{\delta} \to 0 \, ¢³Ð·Õè \delta\to 0.

㹷ӹͧà´ÕÂǡѹ

 f(g(x)+\alpha) - f(g(x)) = \alpha f «Öè§ \frac{\eta(\alpha)}{\alpha} \to 0 \, ¢³Ð·Õè \alpha\to 0. \,

¨Ðä´é

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

«Öè§ \alpha_\delta = \delta g ¨ÐàËç¹ÇèÒ¢³Ð·Õè \delta\to 0 ¹Ñé¹ \frac{\alpha_\delta}{\delta}\to g áÅÐ \frac{\eta(\alpha_\delta)}{\delta}\to 0 ´Ñ§¹Ñé¹

 \frac{f(g(x+\delta))-f(g(x))}{\delta} \to g ¢³Ð·Õè \delta \to 0

[á¡é] ¡®ÅÙ¡â«è¾×é¹°Ò¹

¡®ÅÙ¡â«è¹Ñé¹à»ç¹¤Ø³ÊÁºÑµÔ¾×é¹°Ò¹¢Í§¹ÔÂÒÁ¢Í§Í¹Ø¾Ñ¹¸ì·Ñé§ËÁ´ àªè¹ ¶éÒ E F áÅÐ G à»ç¹ »ÃÔÀÙÁÔºÒ¹Ò¤ (ÃÇÁä»¶Ö§»ÃÔÀÙÁÔÂÙ¤ÅÔ´´éÇÂ) áÅÐ fÿ: EF áÅÐ gÿ: FG à»ç¹¿Ñ§¡ìªÑ¹ áÅжéÒ x à»ç¹ÊÁÒªÔ¡¢Í§ E «Öè§ f ËÒ͹ؾѹ¸ìä´é·Õè x áÅÐ g ËÒ͹ؾѹ¸ìä´é·Õè f(x) áÅéÇ Í¹Ø¾Ñ¹¸ì (͹ؾѹ¸ìà¿Ãવì) ¢Í§¿Ñ§¡ìªÑ¹¤ÍÁâ¾ÊÔµ g o f ·Õè x ¨Ðà»ç¹´Ñ§¹Õé

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

ÊѧࡵÇèÒ͹ؾѹ¸ì¹Õéà»ç¹¡ÒÃá»Å§àªÔ§àÊé¹ äÁèãªèµÑÇàÅ¢ ¶éÒ¡ÒÃá»Å§àªÔ§àÊé¹á·¹´éÇÂàÁ·ÃÔ¡«ì (¨Òâ¤àºÕ¹àÁ·ÃÔ¡«ì) ¡ÒÃÃÇÁ·Ò§´éÒ¹¢ÇҨСÅÒÂà»ç¹¡ÒäٳàÁ·ÃÔ¡«ì

¡ÒáÓ˹´¡®ÅÙ¡â«è·ÕèªÑ´à¨¹ÊÒÁÒö·Óä´é¨Ò¡ÇÔ¸Õ·Õèà»ç¹·ÑèÇä»ÁÒ¡·ÕèÊØ´ ¤×Í ãËé M N áÅÐ P à»ç¹áÁ¹Ôâ¿Å´ì Ck (ËÃ×ͺҹҤáÁ¹Ôâ¿Å´ì) áÅÐãËé

fÿ: MN áÅÐ gÿ: NP

à»ç¹¡ÒÃá»Å§·ÕèËÒ͹ؾѹ¸ìä´é ͹ؾѹ¸ì¢Í§ f á·¹´éÇ df ¨Ðà»ç¹¡ÒÃá»Å§¨Ò¡»ÁÊÑÁ¼Ñʢͧ M ä»Âѧ»ÁÊÑÁ¼Ñʢͧ N áÅÐÊÒÁÒöà¢Õ¹᷹´éÇÂ

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

´éÇÂÇÔ¸Õ¹Õé ÃٻẺ¢Í§Í¹Ø¾Ñ¹¸ìáÅлÁÊÑÁ¼Ñʨж١ÁͧàËç¹ã¹ÃÙ»¿Ñ§¡ìàµÍÃ캹 Category ¢Í§áÁ¹Ôâ¿Å´ì C â´ÂÁÕ¡ÒÃá»Å§ C à»ç¹Êѳ°Ò¹

[á¡é] à·¹à«ÍÃì¡Ñº¡®ÅÙ¡â«è

ÃÙ»ÀÒ¾·Õèà¡ÕèÂÇ¢éͧ

µÔªÁ


µéͧ¡ÒÃãËé¤Ðá¹¹º·¤ÇÒÁ¹Õéè ?

ÊÃéÒ§â´Â :


BuildinGNoomook

ʶҹР: ¼Ùéãªé·ÑèÇä»
¡ÒáèÍÊÃéÒ§