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#1
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͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹¼¡¼Ñ¹
¨Ò¡·Äɮպ·¹Õé¹Ð¤ÃѺ
¶éÒ f à»ç¹¿Ñ§¡ìªÑ¹Ë¹Ö觵èÍ˹Öè§áÅÐËÒ͹ؾѹ¸ìä´é áÅéÇ (f-1)'(x) = 1/f'(x) Åͧ·Ó⨷Âì§èÒÂ æ ¤×Í ¨§ËÒ (f-1)'(x) àÁ×èÍ f(x) = 1/x ¶éÒãªé ·º. ¨Ðä´é -x2 áµè¶éҷӵç æ ¨Ðä´é (f-1)(x) = 1/x ´Ñ§¹Ñé¹ (f-1)'(x) = -1/x2 Why??? ã¤ÃÃÙéªèǺ͡˹èÍÂ
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#2
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¨Ò¡ \(f(f^{-1}(x))=x\) ãªé chain rule ¨Ðä´é \(f'(f^{-1}(x))\cdot(f^{-1})'(x)=1\)
´Ñ§¹Ñé¹¼Á¤Ô´ÇèÒÊٵ÷Õè¶Ù¡¹èÒ¨Ðà»ç¹Íѹ¹Õé¤ÃѺ\[(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}\] |
#3
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¹Õè¤×͵ÑÇ theorem ¢Í§Áѹ¨ÃÔ§æ ¤ÃѺ
ä»à»Ô´Ë¹Ñ§Ê×Í quote ÁÒãËé ä´é´Ñ§¹Õé Let f be 1-1 real-valued function on interval J. Let f be the inverse function for f. If f is continuous at c Î J and if f has a derivative at d=f(c) with f'(d) ¹ 0 then f'(c) exists and f'(c)= 1/ (f'(d))
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#4
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·Õè¡»ÃÐà´ç¹¹Õé¢Öé¹ÁÒà¾ÃÒÐÁÕ¡ÒÃà¢éÒ㨼Դ¡Ñ¹ã¹ËÁÙè¹ÔÊÔµ·ÕèàÃÕ¹á¤Å1 ÇèÒ¡ÒÃËÒ͹ؾѹ¸ì¢Í§ f-1(x) ·Óä´éâ´Â¡ÒÃËÒ͹ؾѹ¸ì¢Í§ f áÅéÇ¡ÅѺàÈÉ¡ÅѺÊèǹ (ÊÙµÃÅÑ´) «Öè§ÇÔ¸Õ¹Õé¹èÒ¨Ðãªéä´é੾ÒÐ linear function Êèǹ¿Ñ§¡ìªÑ¹Í×蹨Ðãªéä´éËÃ×ÍäÁè¹Ñé¹ Íѹ¹Õé¡çäÁèÃÙé¤ÃѺ
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#5
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\[ \frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} \] ·Õ¹ÕéàÃÒÁÒ´ÙÊÙµÃã¹ÍÕ¡ÃÙ»¡Ñ¹ºéÒ§ â´Â·ÑèÇä» ¨Ðà¢Õ¹ÊÙµÃä´éÇèÒ \( \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}\) ËÃ×Í ºÒ§·Õ¡çàËç¹ã¹ÃÙ»¢Í§ \( \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} \) ·Õ¹Õéµéͧ¶ÒÁµÑÇàͧÅèФÃѺÇèÒ y ¤×ÍÍÐäà áÅÐ x ¤×ÍÍÐäà ??? ¡ÒÃãªéÊٵ÷Õè¶Ù¡µéͧ ¹èÒ¨Ðà»ç¹´Ñ§¹Õé¤ÃѺ àÃÔèÁ¨Ò¡ àÃÒµéͧ¡ÒÃ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹¼¡¼Ñ¹ àÃÒ¨Ö§¡Ó˹´ãËé \( y = f^{-1}(x) \) ËÃ×Í¡ÅèÒÇä´éÇèÒ \( x=f(y) \)â´Â¡¯ÅÙ¡â«è¨Ðä´éÇèÒ \( \frac{dy}{dy} = \frac{dy}{dx} \frac{dx}{dy} \) ¨Ðä´éÊٵ÷ÕèàÃÒ¤Ø鹵ҡѹ´Õ¤×Í \[\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} \] ¨Ðä´éÇèÒ \( \frac{dy}{dx} \) ¤×Í͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹ÍÔ¹àÇÍÃìʵÒÁ·ÕèàÃÒµéͧ¡Òà «Öè§ÊÒÁÒöËÒä´é¨Ò¡Êèǹ¡ÅѺ¢Í§ \( \frac{dx}{dy} \) «Öè§ \( \frac{dx}{dy} \) à»ç¹Í¹Ø¾Ñ¹¸ì¢Í§¿Ñ§¡ìªÑ¹·Õèä´é¨Ò¡¡ÒÃà»ÅÕè¹µÑÇá»Ã x,y ÊÅѺ¡Ñ¹ ¢éÍ´Õ¢éÍ¡ÒÃãªéÇÔ¸Õ¹Õé¤×Í àÃÒäÁèµéͧ·Ó¡ÒÃËÒÇèÒ \( f^{-1}(x) \) ÁÕ˹éÒµÒà»ç¹ÍÂèÒ§äÃ(à¾ÃÒкҧ¤ÃÑé§ËÒä´éÂÒ¡) ¡çÊÒÁÒöËÒ͹ؾѹ¸ìä´é·Ñ¹·Õ ÅͧãªéËÅÑ¡¡Ò÷Õè¼ÁÇèҡѺ¿Ñ§¡ìªÑ¹·Õ衵ÑÇÍÂèÒ§Áҵ͹áá¹Ð¤ÃѺ ¨Ò¡ \( f(x) = \frac{1}{x}\) ¡Ó˹´ãËé \( y=f^{-1}(x) \) ËÃ×Í \( x = f(y) = \frac{1}{y} \) ¨Ò¡ÊٵâéÒ§µé¹ \( \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{ - \frac{1}{y^2}} = -y^2 \) «Öè§ àÃÒÊÒÁÒöËÒä´éÇèÒ \( y = f^{-1}(x) = \frac{1}{x}\) á·¹¤èÒŧ仨Ðä´é \( \frac{d}{dx} f^{-1}(x) = - \frac{1}{x^2} \) Åͧãªé¡Ñº¿Ñ§¡ìªÑ¹Í×è¹æ´Ù¤ÃѺ àªè¹ \( f(x) = \sin x \) àÁ×èÍ \( 0<x<\frac{\pi}{2} \) ¡Ó˹´ãËé \( y=f^{-1}(x) \) ËÃ×Í \( x = f(y) = \sin y \) ¨Ò¡ \( \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{\cos y} = \frac{1}{\sqrt{1- \sin ^2 y}} \) «Öè§ àÃÒ·ÃÒºÇèÒ \( \sin y = x \) á·¹¤èÒŧ仨Ðä´é \( \frac{d}{dx} f^{-1}(x) = \frac{1}{\sqrt{1 - x^2}} \) «Öè§à»ç¹ÊٵáÒÃËÒ͹ؾѹ¸ì¢Í§¿Ñ§¡ìªÑ¹ arcsin ·Õèãªé¡Ñ¹ÍÂÙè¹Ñè¹àͧ Åͧãªé¡Ñº¿Ñ§¡ìªÑ¹ Exponential ,logarithm áÅéÇ¡çä´é¼Åàªè¹¡Ñ¹¤Ñº ¶éÒ¼Ô´¶Ù¡ÍÂèÒ§äÃ¡ç ºÍ¡ä´é¹Ð¤ÃѺ Íѹ¹Õéà»ç¹¤ÇÒÁ¤Ô´¢Í§¼Á
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PaTa PatA pAtA Pon! |
#6
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·Õè¤Ø³ m@gpie à¢éÒ㨹Ñ鹶١µéͧáÅéǤÃѺ ·ÕèÁҢͧ¤ÇÒÁà¢éÒ㨼Դ¹èÒ¨ÐÁÒ¨Ò¡àËç¹ÊÙµÃÇèÒ Í¹Ø¾Ñ¹¸ì¢Í§¿Ñ§¡ìªÑ¹¼¡¼Ñ¹ ¤×Í 1/(dy/dx) ¤ÃѺ â´ÂäÁèä´éá·¹¤èÒ¡ÅѺà¢éÒä»ÇèÒ x ¤×ÍÍÐäÃ
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#7
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PaTa PatA pAtA Pon! |
#8
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