21f08-36 - AFA8 8 Óga Ö iØhÑ ×aÒdD eÖ iÚa Ø iÚ e...

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Unformatted text preview: AFA8 8 Óga Ö iØhÑ ×aÒdD eÖ iÚa Ø iÚ e × h ed e Ö iÚa Ø iÚ eÓ fØh eÒa ØÙ Öa ÐÐÓga Ö iØhÑ i×gÙ ÖedÓÙ ØbÝ Øh e Ñ e ØhÓdd e×Ö ib ed iÒ ï5Ù × iÒg Øh e faØØha ØØh e ÐÓga Ö iØhÑ i×Øh e iÒÚ e Ö×eÓ fØh eeÜÔÓÒ eÒ Ø ia Ðea ÐÐÝ ØhÓÙgh iØ× jÙ ×ØØh eha iÒ ÖÙ Ðe :if f ( x ) = ln x Øh eÒ e ln x = x, and so parenleftBig e ln x parenrightBig ′ = ( x ) ′ e ln x (ln x ) ′ = 1 x (ln x ) ′ = 1 , or (ln x ) ′ = 1 x . h eØeÜ Øha ×aÓÙÔ ÐeÓ fÓ Øh e ÖfÓ ÖÑ Ù Ða ×Øha Øa ÖeÖea ÐÐÝ Øh e×aÑ ea ×Øh i×ÓÒ ebÙ ØfaÒ ie Ö ÐÓÓk iÒg × iÒeØh eÝ ×Ø ik Øh eha iÒ ÖÙ Ðe iÒ Øh e Öe : (ln( g ( x )) ′ = g ′ ( x ) g ( x ) , aÒd (ln | x | ) ′ = 1 x . h i× Ða ×ØÓÒ e ÐÓÓk × Ðik eØh e×aÑ e fÓ ÖÑ Ù ÐaeÜeÔ ØÛ iØh Øh eab ×Ó ÐÙ ØeÚa ÐÙ e× igÒB Ù ØÖeÑ eÑ b e ÖØha Ø ln x iØ×e Ðfi×d eÒ edÓÒ ÐÝ fÓ Ö x > ×Ó Øh i× i×aÛ aÝ ØÓeÜ ØeÒd iØØÓ x < ×Ó ÖØÓ fF Ó Ö x < u =- x ×ÓÙ × iÒg Øh eÚ e Ö× iÓÒÛ iØh Øh eha iÒ ÖÙ Ðe : (ln | x | ) ′ = (ln( u )) ′ = u ′ u =- 1 | x | = 1 x , since x < . F Ó ÖÓ Øh e Öba ×e×Øh e fÓ ÖÑ Ù Ða i×× iÑ Ô ÐÝa ÐØe ÖedbÝ Øh eÖÙ Ðe log b ( x ) = ln x ln b (log b ( x )) ′ = 1 x ln b . h i× i×aÒa ÐÓgÓÙ ×ØÓ Øh e fÓ ÖÑ Ù Ða fÓ Ö ( b x ) ′ = b x ln b Û h ihÛ a ×d e Ö iÚ ed ea Ö Ðie Ö E ÜaÑ Ô Ðe × : AFA8 d dx ln parenleftbigg x + 1 √ x- 2 parenrightbigg = d dx ( ln ( x + 1)- ln ( √ x- 2 )) = d dx parenleftbigg...
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This note was uploaded on 11/24/2011 for the course MATH 21 taught by Professor Stanley during the Fall '09 term at Lehigh University .

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21f08-36 - AFA8 8 Óga Ö iØhÑ ×aÒdD eÖ iÚa Ø iÚ e...

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