n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)

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1 8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r Theorem 6.. (.) i 6.. was as follows. a f <> 0 g 0 d m - a - m - - s0 - + r s - f ( -r ) g r (.) r f < + r> g r f < -r+ s> a g s -r + (-) m - r- r- s0ts Ct s C r (-) m - + (,m) k0 Should be oted here is the et two. a -r a a-r+ d r m+--r+ t m- f m+ -r+s a g ( m+ s) -r a -r aa r+ -C m+k k a f < m+ k > g ( m+ k) d a i Whe of the rd lie does ot eist, whe 0 of the d lie does ot eist also. ii Whe the biomial coefficiet of the th lie is geeralized, the upper limit - of ca be replaced by. Sice >0 - at the time 0,,,, if the ide of the itegratio operator is substituted for - i cosideratio of these, it becomes as follows. a f <> 0 g 0 d - m - a (-) m + ( -,m) k0 r m+k f < -+ r> g r - - k a Sice m may be arbitrary iteger, whe m +, it is as follows. a f <> 0 g 0 d - a f < -+ r> g r r (-) ( -,+) k0 ++k k a f < > a m+ k g ( m+ k) d - ++ k g ( ++ k) d - a f < > However, sice ( -,+) for 0,,,, the d lie disappears. That is, a f g d - f < -+ r> g r 0,,, a r The, replacig the itegratio operators d -,<- + r > with the differetiatio operators,( -r) respectivly, we obtai the desired epressio. d r - -

2 8. Higher Derivative of ^a f Formula 8..0 Whe z deotes the gamma fuctio ad f is times differetiable cotiuous fuctio, the followig epressios hold for a atural umber. f ( + ) r ( + -r) - r f ( - r ) (0.) Where, if -,-,-,, it shall read as follows. ( +) (-) -r ( - +r) + -r ( -) Especially, whe m 0,,, m m f ( +m ) m- r f ( - r ) (0.') r ( +m-r) Whe -,-,-, & - -,-,-, f ( +) -+ r f r (0.) r ( + - +r) Whe g( ) i Theorem 8.., sice r ( + ) - r ( + -r) we obtai the followig epressio immediately. f ( + ) r ( + -r) Especially, whe m 0,,,, (0.) is as follows. m f m ( +m ) r ( +m-r) ( +m ) r ( +m-r) We adopt the coveiet latter for mathematical software. - r f ( - r ) (0.) m- r f ( - r ) m- r f ( - r ) ( +m ) 0 for m<r ( +m-r) Whe -,-,-,, from..5 ( Properties of the Gamma Fuctio ) (5.5), ( -z) (-) - ( +z+) ( is a o-egative iteger ) ( -z-) ( +z) The substitutig -z +, r for this, we obtai the proviso. Last, replacig r with -r i (0.), we obtai (0.). r 0 for <r m Below, substitutig various fuctios f for Formula 8..0, we obtai various formulas. Although there are ad i Formula 8..0, sice is almost meaigless i the case of higher differetiatio, we adopt i priciple. - -

3 8.. Higher Derivative of ( a+b) p ( c+d) q Formula 8.. The followig epressios hold for p >0 ad,,,. ( a+b) p ( c+d) q ( /a) -+ r ( +p) ( +q) r ( /c) r ( +p - +r) ( +q -r) Especially, whe m 0,,, ( a+b) p ( c+d) m m ( /a) -+ r ( +p) ( +m) r ( /c) r ( +p - +r) ( +m-r) Let f( ) ( a+b) p, g( ) ( c+d) q, the - r ( a+b) p f r ( c+d) q g -r r c a -+r ( +p) ( +p - +r) -r ( +q) ( +q -r) Substitutig these for Theorem 8.., we obtai (.). ( c+d) q-r Ad especially, whe q m 0,,,, from (.) ( a+b) p ( c+d) m ( /a) -+ r ( +p) ( +m) r ( /c) r ( +p - +r) ( +m-r) m ( /a) -+ r ( +p) ( +m) r ( /c) r ( +p - +r) ( +m-r) We adopt the latter epressio as (.'). ( a+b) p-+r ( c+d) r-q (.) ( a+b) p-+ r ( c+d) r-m ( a+b) p-+r (.') q,,, ( a+b) p-+r ( c+d) r-m ( a+b) p-+r ( c+d) r-m Eample The d order derivative of - + Substitutig a, b-, p/, c, d, q/, for (.), - + r - r- r ( /) ( /) - r ( +) ( r-/) ( /-r) ( -) - + ( - )( +) ( +) Eample' The rd order derivative of - ( +) Substitutig a, b-, p/, c, d, m, for (.'), - -

4 - ( +) r -( +) r ( /) ( -/+r) ( -r) 8( -) ( -) r ( +) r- - + ( + )( -) + ( -) Eample The rd order derivative of - / ( +) Whe q -,-,-,, (.) ca be read as follows. ( a+b) p ( c+d) q ( /a) -+ r ( +p) ( -q +r) r (-/c) r ( +p - +r) ( -q) Substitutig a, b-, p/, c, d, q-, for this, ( a+b) p-+ r ( c+d) r-q r (-) r ( /) ( +r) ( -/+r) 8( -) 9 + ( -) ( +) ( -) r ( +) r+ 7 + ( -)( +) 6 - ( +) 8.. Higher Derivative of log Formula 8.. log - - (-) -r ( -r) ( +) r - ( +) + - log ( + -r) ( + -) Especially, whe m 0,,, m log - - (-) -r ( -r) ( +m) r m- ( +m) + m- log ( +m-r) ( +m-) Where, there shall be o d term of the right side at the time of m<. (.) (.') Let f( ) log. The ( log ) -(-) ( -r) -+r r 0,,,- log r Substitutig these for (0.) i Theorem 8..0, we obtai (.). Whe m 0,,,, applyig (0.'), we obtai the followig. m log - - (-) -r ( -r) ( +m) r m- ( +m) + m- log ( +m-r) ( +m-) m + ( +m) m- ( log ) ( -r) r ( +m-r) r + Where, sice r does ot reach at the time of m<, the d term does ot eist. - -

5 Eample The rd order derivaive of Substitutig /, for (.), log - (-) -r log 5 5 ( -r) ( /) - ( /) - + log r ( /-r) ( -/) 0!! 0! - - ( / ) ( /) ( /) + / - / -/ Eample' The d order derivaive of log 5 ( /) - + log ( -/) 5 5 ( /) log ( -/) - + log 8 Substitutig m, for (.'), log - (-) -r ( -r) r + log ( -r) - (-)!! + (-) 0! 0!!!! + log! ( log ) Eample" The rd order derivaive of log Substitutig m, for (.'), log - (-) -r ( -r) - r ( -r) 0 - -! + -! + - Eample The rd order derivaive of log / -! Whe -,-,-,, (.) ca be read as follows. log -(-) - ( -r) ( - +r) - + (-) ( - +) - log r ( -) ( -) Substitutig -, for this, log -(-) - ( -r) ( +r) - + (-) - - log r - 0 ( + ) ( + ) ( - ) 6 log ( - 6log ) - 5 -

6 8.. Higher Derivatives of si, cos Formula 8.. si {} cos {} r ( +) -r ( + -r) si+ r ( +) -r ( + -r) cos+ ( -r) ( -r) Especially, whe m 0,,, m si {} m ( +m) m-r ( -r) r ( +m-r) si+ m cos {} m ( +m) m-r ( -r) r ( +m-r) cos+ (.s) (.c) (.'s) (.'c) Eample The d order derivative of si Substitutig /, for (.s), si ( /) -rsi r /-r + ( -r) / si + + / - si 0 / / + - si + - cos - 9-5si 5 / - si -/ + Eample' The rd order derivative of si Substitutig m, for (.'s), si {} -r r -r si+ si cos - 6 si + 6 cos 8.. Higher Derivatives of sih, cosh ( -r) + + si + 0 si + Formula 8.. sih cosh r ( + ) ( + -r) r ( + ) ( + -r) - r e -(-) -( -r) e - - r e +( -) -( -r) e - (.s) (.c) - 6 -

7 Especially, whe m 0,,, m sih m m cosh m ( +m ) r ( +m-r) ( +m ) r ( +m-r) m- r e -(-) -( -r) e - m- r e +( -) -( -r) e - (.'s) (.'c) Eample The d order derivaive of sih Substitutig /, for (.s), sih ( /) r /-r -r e -(-) -( -r) e - ( /) sih + ( /) - cosh + 5 ( /) - sih 0 / / -/ sih + - cosh - 9-5sih Eample' The rd order derivative of sih Substitutig m, for (.'s), sih {} r -r 0 cosh + -r e -(-) -( -r) e - cosh + 6 sih + 6 cosh sih + 0 cosh - 7 -

8 8. Higher Derivative of log f 8.. Higher Derivative of ( log ) Formula 8.. log ( log ) - r -r r r (.) Let f( ) g( ) log. The ( log ) ( - r ) (-) - r- ( -r) - r, ( log ) r (-) r- ( r) r,, Substitutig these for Theorem 8.., log ( log ) ( - r ) ( log ) r r 0 ( log ) ( log ) r ( log ) ( - r ) ( log ) r r ( log ) ( - ) ( log ) (-) - log + + (-) - r ( -r) r r log - r -r r Eample The rd order derivative of ( log ) log - - log - r -r r log - ( )- ( log -6) r r 8.. Higher Derivatives of log si, log cos Formula 8.. ( log si ) log si+ r + (-) r r- r si r + ( -r) (.0s) - 8 -

9 ( log cos ) log cos+ r + (-) r r- r cos r + ( -r) (.0c) Eample The rd order derivative of log si ( log si ) log si+ -log cos + + r si + - r- r r si + r -log cos - si - cos + si - + ( -r) si + 0 si Higher Derivatives of log sih, log cosh Formula 8.. log sih log log cosh log e -(-) - e - + (-) r r- r e +( -) - e - + (-) r r- r Eample The th order derivative of log cosh ( log cosh ) log e +( -) - e - + r r r r r - r- ( r) r r e -(-) r- e - (.0s) e +( -) r- e - (.0c) e +( -) r- e - log cosh + sih - cosh + sih - cosh log cos + sih - cosh + sih - cosh - 9 -

10 8. Higher Derivative of e^ f 8.. Higher Derivative of e Formula 8.. e e ( + ) - r r ( + -r) for -,-,-, (.) e ( - +r) - r r ( -) for -,-,-, (.) - r Especially, whe m 0,,, e m e m ( +m ) r ( +m-r) m- r Substite f( ) e for Theorem The sice e ( -r ) e, we obtai the desired epressio immediately. Eample The d order derivative of e e e ( /) r /-r (.') - r e ( /) + ( /) - + ( /) - 0 / / -/ e e + - Eample The d order derivative of e / e r e ( +r) (-) r - r e e Higher Derivative of e log Formula 8.. e log e log + e (-) r r- r Let f( ) e, g( ) log. The r r (.) - 0 -

11 ( log ) r (-) r- r r r,,, Substitutig this for Theorem 8.., e log e ( log ) r r 0 e e log + e (-) r r- r r r log 0 + e r log r Eample The th order derivative of e log e log e log + e (-) r r- r r r e log + e ( ) e log + e Higher Derivatives of e si, e cos Formula 8.. e si e cos si - e si + - si e cos+ (.0s) (.0c) " 共立数学公式 " p87 was posted as it was. Eample e si e cos si - e si + - si e cos+ e cos -e si + cos Higher Derivatives of e si, e cos ed ow. There is o ecessity for Theorem 8... However, darig use Theorem 8.., we obtai a iterestig result. Trigoometric Polyomial Formula 8..' r r si + si - si + (.s) - -

12 r Especially, whe 0 r r r cos+ - si cos+ r si si - si r cos si - cos (.c) (.'s) (.'c) Substitutig f( ) e, g( ) si, cos for Theorem 8.., e si e r si r + e cos e r r cos+ Ad comparig these with Formula 8.., we obtai the desired epressios. Whe 5, if both sides of (.s) are illustrated, it is as follows. Both overlap eactly ad blue (left) ca ot be see. Alterative Biomial Polyomial r r Removig si, cos from (.'s), (.'c), we obtai the followig iterestig polyomial. Formula 8.." Whe deotes the floor fuctio, the followig epressios hold. ( -)/ / r+ - r r - r si cos (.s) (.c) - -

13 Sice the odd-umbered terms of the left side i (.'s) are all 0, r Also, sice si r 0 si 0 + si + si + si + 5 si si/ - / i the right side i (.'s), ( -)/ r+ si - r Net, sice the eve-umbered terms of the left side i (.'c) are all 0, cos cos r Also, sice r 0 cos 0 cos cos + + si ++ si si ( -)/ - r+ cos + - / si/ - / i the right side i (.'c), / cos r - r I additio, this formula is kow. (See " 岩波数学公式 Ⅱ" p) Note Whe k - Eample ( -)/, k,,, / r+ - r r - r si Higher Derivatives of e sih, e cosh Formula 8.. e sih e r cos (.s) cos + / cos / 5 cos - / r - r (.c) e - (-) -r e - (.0s) r - -

14 e cosh e Eample Note e sih 0 e e cosh e 0 e + (-) -r e - (.0c) r 0 e - (-) -r e - e r sih - -r e - r e + e 0 sih + cosh + sih + cosh e ( sih + cosh ) The followig formula is kow for a atural umber. e sih e cosh - e ( sih + cosh ) However, this formula does ot hold for 0. That is, i this formula, the atural umber is ietesible to the real umber p. So, this is isufficiet as a geeral formula. - -

15 8.5 Higher Derivative of f / e^ 8.5. Higher Derivative of e - Formula 8.5. e - e - - -( -r) (-) - e - Especially, whe m 0,,, e - m e - m - -( -r) ( + ) - r for -,-,-, (.) r ( + -r) ( - +r) - r for -,-,-, (.) r ( -) ( +m ) r ( +m-r) m- r Substite f( ) e - for Theorem The sice e - ( -r ) (-) -( -r) e -, we obtai the desired epressio immediately. (.') Eample The rd order derivative of e - / e - (-) - e - 0 ( +r) --r r -e e Eample' The rd order derivative of e - 7 e - 7 e - 7 -r ( 8 ) r ( 8-r) 7 7- r e ( 8) 5 + ( 8) 6 5 e Higher Derivative of e - log Formula 8.5. e - log - - r log - r r r e (.) Let f( ) e -, g( ) log. The - 5 -

16 e - ( -r ) (-) - +r e - r ( log ) r (-) r- r r,,, Substitutig these for Theorem 8.., e - log (-) - +r e - ( log ) r r 0 (-) - e - ( log ) 0 + r (-) - r e log - r r r Eample The th order derivative of e - log e log - - r log - r e log - e e log e - e 8.5. Higher Derivatives of e - si, e - cos r r (-) - +r e - ( log ) r r + ( ) + ( ) + r Formula 8.5. e - si e - cos -si - e - si - - -si e - cos- (.0s) (.0c) Replacig with - i Formula 8.., we obtai the desired epressios. Eample e - si e - cos -si - e - si - - -si e - cos- -e - cos -e - si - cos Higher Derivatives of e - sih, e - cosh ed ow. There is o ecessity for Theorem 8... Darig use Theorem 8.., we obtai the followig epressio first. -r e - si e - ( - ) r si r

17 Ad from this ad (.0s), we obtai -r ( - ) r si r + si - si - A similar epressio is obtaied about e - r r cos too. The removig si, cos from these, we obtai the completely same results as Formula 8.." Higher Derivatives of e - sih, e - cosh Formula 8.5. e - sih e - e - cosh e r - -+ r e - (-) e - (.0s) r e + (-) e - (.0c) r Substitutig f( ) e -, g( ) sih, cosh for Theorem 8.., we obtai the dsired epressios. Eample Note e - sih 0 e r 0 e - (-) -r e - e - sih r e - cosh e - (-) -+ r e r + (-) -r e cosh + sih - cosh + sih e - -e - cosh - sih The followig formula is kow for a atural umber. e - sih e - cosh (-) - e ( cosh -sih ) However, this formula does ot hold for 0. That is, i this formula, the atural umber is ietesible to the real umber p. So, this is isufficiet as a geeral formula

18 8.6 Higher Derivatives of si f, cos f 8.6. Higher Derivatives of si, cos Formula 8.6. si - cos - + cos cos - + (.0s) (.0c) From Formula 8.6.' metioed et, Here cos Usig this, Ad sice cosa cosb cos r r cos+ cos( A+B ) +cos( A-B) cos -r cos + r r + + cos -r cos +, r - r r 0 + cos ( -) r r substitutig these for the above, we obtai (.0c). (.0s) is also obtaied i a similar way. Eample si - cos - + cos cos - + cos cos - si si 8 si cos Formula 8.6.' si cos r r si+ cos+ -r si + r -r cos + r (.s) (.c) Substitutig f( ) g( ) si for Theorem 8.., we obtai (.s). (.c) is also obtaied i a similar way

19 Formula 8.6." Whe deotes the floor fuctio, the followig epressios hold. / r ( -)/ - (.e) r+ - (.o) (.s) is trasformed as follows. i.e. si / r ( -)/ + / si+ r r - r si+ ( -)/ + -r si + r r -r si + r+ si ( -r-) ( r+) + si + si+ - r si si+ si ( -r) si ( -r-) r+ si + / r O the other had, (.0s) is trasformed as follows too. cos ( -)/ - cos+ cos si si - + si - cos - + cos From these, the followig epressio follows. si + si / r - - cos + cos I order to hold this equatio for arbitrary, the followigs are ecessary. / r ( -)/ - - 0, r I additio, this formula is kow. (See " 岩波数学公式 Ⅱ" p) ( -)/ r+ r Higher Derivatives of si, cos Formula 8.6. si cos si + cos + - si+ + cos+ (.0s) (.0c) - 9 -

20 From Formula 8.6.' metioed et, it is obtaied i a similar way i the case of the d degree. However, it is ot so easy as the case of the d degree. ( See 0.. ) Eample si cos si + cos si+ - si + si cos+ 7 si + si Formula 8.6.' si si 0 si+ - r r cos r- + cos cos 0 cos+ + r r cos r- + ( -r) r si + ( -r) r cos + (.s) (.c) Substitutig f( ) si, g( ) si for Theorem 8.., we obtai (.s). (.c) is also obtaied i a similar way. Formula 8.6." / r- ( -)/ r From (.s) r si si r+ - + (-) (.e) 0 si + si 0 si+ ( -)/ - / - r r - (.o) - r r r- cos + r+ cos r + cos r- + ( -r) r si + r+ ( -r-) si + r si + -r - 0 -

21 Whe, si - - cos si+ ( -)/ ( -r-) + r+ r si si+ - r r - r ( -)/ - / - ( r ) r- ( -r) cos si+ r+ r si cos+ / 0 - cos si+ - r r ( -)/ - r+ si cos+ / - r- cos r si+ si + - si + si + si + si + si + si + si + / r- ( - )/ r Whe, si r - cos si cos cos + si si r- cos si+ 0 - si cos + + si si - ( cos cos - si si ) + si si - cos - ( cos - cos ) - cos - si+ - +( -) r (-) + si + - cos si+ - si cos

22 / r- ( - )/ r si + si + si + si + r + 5 cos si + si cos + ( si - si ) + si + 9 si - si ( -) + r+ - -(-) Hereafter, by iductio, we obtai the desired epressios. If both sides of Formula8.6." are illustrated, it is as follows. The left side is blue lie ad the right side is red poit. - -

23 8.6. Higher Derivatives of the product of trigoometric ad hyperbolic fuctios Formula 8.6. ( sisih ) ( sicosh ) r r ( cossih ) ( coscosh ) r r si+ si+ cos+ cos+ ( -r) ( -r) ( -r) ( -r) Substitutig f( ) si, g( ) sih for Theorem 8.., we obtai (.). The others are also obtaied i a similar way. Eample ( sicosh ) r r ( coscosh ) si+ cos+ ( 0-r) ( -r) e -(-) -r e - (.) e +( -) -r e - (.) e -(-) -r e - (.) e +( -) -r e - (.) e +( -) -0 e - sicosh e +( -) -r e - 0 cosh +cos sih +cos cos+ + + cosh -coscosh - si sih + coscosh -si sih - -

24 8.7 Higher Derivatives of sih f, cosh f 8.7. Higher Derivatives of sih, cosh Formula 8.7. sih cosh e -(-) -+r e - e -(-) -r e - (.s) r e +( -) -+r e - e +( -) -r e - (.c) r Substitutig f( ) g( ) sih for Theorem 8.., we obtai (.s). (.c) is also obtaied i a similar way. Eample sih e -(-) -0+r e - e -(-) -r e - sih r cosh e +( -) -+r e - e +( -) -r e - r 0 sihcosh + coshsih + sihcosh + coshsih 8sihcosh sih Alie's Mathematics K. Koo - -