ke kx 1 cosx sin kx k cos kx sin x k sin kx tan x = sin x sec 2 x tan kx k sec 2 kx cosec x = 1 cosec x cot x sec x = 1
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3 8. Tble of derivtives Introduction This leflet provides tble of common functions nd their derivtives.. The tble of derivtives d = f( ( d k, n constnt n, n constnt n n n e e k e ke k ln = log e sin cos sin k k cos k cos sin cosk k sin k tn = sin cos sec tn k k sec k cosec = sin cosec cot sec = cos sec tn cot = cos cosec sin sin cos tn + cosh sinh sinh cosh tnh sech sech sech tnh cosech cosechcoth coth cosech cosh sinh + tnh c Person Eduction Ltd 000
4 Eercises. In ech cse, use the tble of derivtives to write down d d. = 8 b = c = 0 d = e = 5 f = 7 g = 3 h = / i = / j = sin k = cos l = sin 4 m = cos n = e 4 o = e p = e q = e r = ln s = log e t = u = 3 v = w = e /. You should be ble to use the tble when other vribles re used. Find d dt if = e 7t, b = t 4, c = t, d = sin 3t. Answers. 0, b 0, c 0, d, e 5 4, f 7 6, g 3 4, h /, i 3/, j cos, k sin, l 4 cos 4, m sin, n 4e4, o e, p e, q e, r, s t / = / =, u 3 /3 = 3 /3 = 3 3, v 3/, w e/.. 7e 7t, b 4t 3, c t, d 3 cos3t c Person Eduction Ltd 000
5 Tble of Derivtives Throughout this tble, nd b re constnts, independent of. F ( f( + bg( f( + g( f( g( f( f(g( f(g(h( f( g( g( f ( g( F ( = df d f ( + bg ( f ( + g ( f ( g ( f ( f (g( + f(g ( f (g(h( + f(g (h( + f(g(h ( f (g(f(g ( g( g ( g( f ( g( g ( 0 0 g( sin sin g( cos cos g( tn csc sec cot e e g( ln g( g ( cos g ( cos g( sin g ( sin g( sec csc cot sec tn csc e g (e g( (ln g ln g( ( g( log ln rcsin rcsin g( g ( g( rccos rctn + g rctn g( ( +g( rccsc rcsec rccot +
6 Tble of Indefinite Integrls Throughout this tble, nd b re given constnts, independent of f( f( + bg( f( + g( f( g( f( u(v ( f ( ( ( nd C is n rbitrr constnt. F ( = f( d f( d + b g( d + C f( d + g( d + C f( d g( d + C f( d + C u(v( u (v( d + C F ( ( where F ( = f( d + C + C g( g ( sin g ( sin g( cos tn csc sec cot sec csc sec tn csc cot e C if ln + C g( C if cos + C cos g( + C sin + C ln sec + C ln csc cot + C ln sec + tn + C ln sin + C tn + C cot + C sec + C csc + C e + C e g( g ( e g( + C e e + C ln + C ln ln + C g ( g( rcsin + C rcsin g( + C rcsin + C + rctn + C g ( +g( + rctn g( + C rctn + C rcsec + C
7 Properties of Eponentils In the following, nd re rbitrr rel numbers, nd b re rbitrr constnts tht re strictl bigger thn zero nd e is , to ten deciml plces. e 0 =, 0 = e + = e e, + = 3 e = e, = 4 ( e = e, ( = 5 d d e = e d, d eg( = g (e g( d, d = (ln 6 e d = e + C, e d = e + C if 0 7 lim e =, lim e = 0 lim =, lim = 0 if > lim = 0, lim = if 0 < < 8 The grph of is given below. The grph of, for n >, is similr. =
8 Properties of Logrithms In the following, nd re rbitrr rel numbers tht re strictl bigger thn 0, is n rbitrr constnt tht is strictl bigger thn one nd e is , to ten deciml plces. e ln =, log =, log e = ln, log = ln ln log ( =, ln ( e = ln = 0, log = 0 ln e =, log = 3 ln( = ln + ln, log ( = log + log 4 ln ( = ln ln, log ( = log log ln ( = ln, log ( = log, 5 ln( = ln, log ( = log 6 d d ln =, d d ln(g( = g ( g(, d d log = ln 7 d = ln + C, ln d = ln + C 8 lim ln =, lim ln = 0 lim log =, lim log = 0 9 The grph of ln is given below. The grph of log, for n >, is similr..5.0 = ln
9 Bsic Differentition Formuls In the tble below,? œ 0 ÐBÑ œ ÐBÑ represent differentible functions of B.- Derivtive of constnt œ!. w w Derivtive of constnt (-? œ - ÐWe could lso write Ð-0Ñ œ -0, nd could use multiple the prime notion in the other formuls s well Derivtive of sum or. œ Product Rule (?@Ñœ?.@.? Quotient Rule œ Chin Rule.C.C œ. 8 8 ". 8 8 " B œ 8B? œ 8?. B B.?? + œ (ln œ (ln + +. B B.?? (If + = / / œ / / œ /.. log + B œ (ln + B log +? œ (ln +?.. B? (If +œ/ ln B œ ln? œ. sin cos. sin cos Bœ B?œ?. cos sin. cos sin Bœ B?œ?. #. # tn Bœsec B tn?œ sec?. #. # cot Bœ csc B cot?œ csc?. sec sec tn. sec sec tn Bœ B B?œ??. csc csc cot. csc csc cot Bœ B B?œ??. ". " sin Bœ sin?œ. rcsin. rcsin B œ È" B? œ # È"? #. ". " tn Bœ tn?œ. rctn =. rctn = B " B#? "?#
10 Bsic Forms n d = n + n+ ( d = ln ( udv = uv vdu (3 + b d = ln + b (4 Integrls of Rtionl Functions ( + n d = ( + d = + (5 ( + n+, n (6 n + ( + n d = ( + n+ ((n + (n + (n + (7 + d = tn (8 + d = tn (9 + d = ln + (0 + d = tn ( 3 + d = ln + ( + b + c d = + b 4c b tn 4c b (3 ( + ( + b d = b ln + b +, b (4 ( + d = + ln + (5 + + b + c d = ln + b + c b + b 4c b tn 4c b Integrls with Roots (6 d = 3 ( 3/ (7 d = ± (8 ± d = (9 d = 3 ( 3/ + 5 ( 5/ (0 ( b + bd = b 3 ( ( + b 3/ d = 5 ( + b5/ ( d = ± 3 ( ± (3 ( d = ( tn (4 + d = ( + ln [ + + ] (5 Tble of Integrls + bd = 5 (b + b b (6 ( [ + bd = ( + b ( + b 4 3/ b ln + ] ( + b 3 ( + bd = (7 [ b b 8 + ] 3 ( + b 3 + b3 8 5/ ln + ( + b (8 ± d = ± ± ln + ± (9 d = + tn (30 ± d = 3 ( ± 3/ (3 ± d = ln + ± (3 d = sin (33 ± d = ± (34 d = (35 ± d = ± ln + ± (36 + b + cd = b + + b + c 4 4c b + ln + b + ( 8 + b + c (37 3/ + b + c = ( 48 + b + c 5/ ( 3b + b + 8(c + +3(b 3 4bc ln b b + c (38 + b + c d = ln + b + ( + b + c + b + c d = + b + c (39 b ln + b + ( + b + c (40 3/ d ( + = 3/ (4 + Integrls with Logrithms ln( + bd = ln d = ln (4 ln d = (ln (43 ( + b ln( + b, 0 (44 ln( + d = ln( + + tn (45 ln( d = ln( + ln + (46 ln ( + b + c d = 4c b tn + b 4c b ( b + + ln ( + b + c (47 ln( + bd = b 4 + ( b ln( + b (48 ln ( b d = + ( ln ( b (49 b Integrls with Eponentils e d = e (50 e d = e + i π erf ( i, 3/ where erf( = e t dt (5 π e d = ( e (5 e d = 0 ( e (53 e d = ( + e (54 ( e d = + 3 e (55 3 e d = ( e (56 n e d = n e n n e d (57 n e d = (n Γ[ + n, ], n+ where Γ(, = t e t dt (58 e d = i π erf ( i (59 π e d = erf ( (60 e d = e (6 e d = 4 π erf( 3 e (6 04. From lst revised June 4, 04. This mteril is provided s is without wrrnt or representtion bout the ccurc, correctness or suitbilit of the mteril for n purpose, nd is licensed under the Cretive Commons Attribution-Noncommercil-ShreAlike 3.0 United Sttes License. To view cop of this license, visit or send letter to Cretive Commons, 7 Second Street, Suite 300, Sn Frncisco, Cliforni, 9405, USA.
11 Integrls with Trigonometric Functions sin d = cos (63 sin d = sin 4 sin n d = [ cos F, n, 3 ], cos (64 (65 sin 3 3 cos cos 3 d = + (66 4 cos d = sin (67 cos d = + sin 4 cos p d = ( + p cos+p cos sin bd = F [ + p,, 3 + p, cos cos 3 d = 3 sin 4 cos[( b] ( b + sin 3 sin sin[( b] cos bd = 4( b + cos sin bd = sin b b ] (68 (69 (70 cos[( + b], b ( + b (7 sin[( + b] 4( + b (7 sin cos d = 3 sin3 (73 cos[( b] 4( b cos[( + b] 4( + b cos b b (74 cos sin d = 3 cos3 (75 sin cos bd = 4 + sin 8 sin b 8b sin cos d = 8 sin[( b] 6( b sin[( + b] 6( + b sin 4 3 (76 (77 tn d = ln cos (78 tn d = + tn (79 sec 3 d = sec tn + ln sec + tn (84 sec tn d = sec (85 sec tn d = sec (86 sec n tn d = n secn, n 0 (87 csc d = ln tn = ln csc cot + C (88 csc d = cot (89 csc 3 d = cot csc + ln csc cot (90 csc n cot d = n cscn, n 0 (9 sec csc d = ln tn (9 Products of Trigonometric Functions nd Monomils cos d = cos + sin (93 cos d = cos + sin (94 cos d = cos + ( sin (95 cos d = cos + 3 sin (96 n cosd = (in+ [Γ(n +, i +( n Γ(n +, i] (97 n cosd = (in [( n Γ(n +, i Γ(n +, i] (98 sin d = cos + sin (99 cos sin sin d = + (00 sin d = ( cos + sin (0 sin d = cos + 3 sin (0 e b cos d = e cos d = e (sin + cos (06 + b eb ( sin + b cos (07 e sin d = e (cos cos + sin (08 e cos d = e ( cos sin + sin (09 Integrls of Hperbolic Functions cosh d = sinh (0 e cosh bd = e [ cosh b b sinh b] b b e 4 + ( = b sinh d = cosh ( e sinh bd = e [b cosh b + sinh b] b e 4 b = b (3 e tnh bd = e (+b [ ( + b F + b,, + b, eb] [ e F,, E, eb] b (4 b e tn [e ] = b tnh d = ln cosh (5 cos cosh bd = cos sinh bd = sin cosh bd = [ sin cosh b + b +b cos sinh b] (6 [b cos cosh b+ + b sin sinh b] (7 [ cos cosh b+ + b b sin sinh b] (8 tn n d = tnn+ ( + n F ( n + tn 3 d =,, n + 3, tn (80 ln cos + sec (8 sec d = ln sec + tn = tnh ( tn (8 sec d = tn (83 n sin d = (in [Γ(n +, i ( n Γ(n +, i] Products of Trigonometric Functions nd Eponentils e b sin d = (03 e sin d = e (sin cos (04 + b eb (b sin cos (05 sin sinh bd = [b cosh b sin + b cos sinh b] (9 sinh cosh d = [ + sinh ] (0 4 sinh cosh bd = [b cosh b sinh b cosh sinh b] (
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