08 No. : No. : No.3 : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 : No. :
sin cos No. sine, cosine : trigonometric function π : π = 3.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even. sin. sin 3. sin + π sin cos Pythagoras sin + cos = cotangent : cot = cos sin = tan secant : sec = cos cosecant : cosec = sin co 7 tan cot sin cosec cos sec e : eponential function 3 = 8 3 eponent y = e : = ep : e =.78 : 8 y = ep y = ep addition formulas sin α ± β = sin α cos β ± cos α sin β cos α ± β = cos α cos β sin α sin β 3 tan 4 y = sin y = cos 5 sinh cosh π 0 sin d π 0 cos d cosh sinh = 6 sin 3 = 4 sin 3 + 3 4 sin sin 4 = 8 cos 4 cos + 3 8 cos 3, cos 4 tangent : tan = sin cos sinh = cosh = tanh = hyperbolic function e e : e +e : sinh cosh = e e e +e : cosh sinh sinh n sinh n 9 sinh cosh tanh? = 0? sinh α ± β = sinh α cosh β ± cosh α sinh β cosh α ± β = cosh α cosh β ± sinh α sinh β 0 cosh sinh =, sinh α ± β =, cosh α ± β =
No. log logarithmic function. y = e < <, y > 0 = log y y > 0, < < inverse trigonometric functions II = arcsin y y = sin π π, y = arccos y y = cos 0 π, y = arctan y y = tan π < < π, < y < [ ], f f f f =, f f = e log = > 0, log e = < < base a> 0, y = a = log a y e log, ln 0 log ln,,, log ln log e.,, sin y Sin y sin y, arcsin y. arc.,.,. Sin sin π 6 = arcsin 3 = arccos = arctan 3 π 4 = arcsin = arccos = arctan π 3 = arcsin 3 = arccos = arctan 3 a, b, c,, y > 0, a, c log a y = log a + log a y log a y = y log a log a b = log c b log c a a a y = a +y, a y = a y., e. a = e log e a log a = log e log e a. a > 0, a, b > 0. log a. log a a 3. log a a 4. log /a b log a b 5. log b a log a b 6. log a b log a b 7. log a b loga b 8. log 8 9. log 8 6 0. log 7 8,,.,,. 3.. arcsin. arcsin 3. arccos 4. arccos 5. arctan 6. arctan 3 7. sin arccos 8. cos arcsin
No.3 sin arccos 3 5 y = sinh = arcsinh y cos arctan y = cosh = arccosh y 3 tan arcsin y = tanh = arctanh y 0 log : 4 arcsin = arctan arcsinh = log + +, < < 5 arccos = arcsin 4 arccosh = log ±, 6 6 arctan = arccos = ± log + 6 7 cos arcsin 3 arcsin arctanh = 4 log +, < < C 8 arcsin + arcsin 3 = arcsin [ π, π ] 3 arcsinh arccosh arctanh 9 arctan + arctan 3 0 arcsin 3 + arcsin 3 3 sin arcsin 4 cos arccos 6 3 arcsin 3...................... 4 arcsin.................. 3 5 arccos...................... 6 arccos................. 7 arctan 3..................... 8 arctan 3................. 9 sin arctan 3 4............. 0 cos arcsin 3............. tan arccos 3 3.......... arccos 3 5 arccos 4 5 = arccos..!. 4 differential:, differentiation:, derivative: f = df d = lim f + h f h 0 h. f + g = f + g. fg = f g + fg fgh = f gh + fg h + fgh f f f 3 f 4 = + + + 3. f g = f g fg g 4. chain rule df g = df dg d dg d f g = f g g 5. dy d = f = d dy f f 3 4 f/g = fhg, hy = y
No.4 derivatives of basic functions. s = s s. e = e 3. log = log = 4. sin = cos, cos = sin 5. tan = cos, cot = sin 6. arcsin = 7. arccos = 8. arctan = + 9. sinh = cosh, cosh = sinh 0. tanh =. arcsinh = cosh, coth = sinh +. arccosh = ± 3. arctanh = 9 9 4 f = f g fg, g g 4 5 cos + sin = 9 0 cosh sinh = dy d = d dy 3 3 4 4 6 7 5 5 8 6 9 3 7 differentiation of composite functions 8. 3. 4 cos 5 6 3. sin arcsin 3 4. arccos cos 5. arctan sin { 6. ep } 5 3 7. log 4 + 3 + + 5 8. 3 3 3.................... 9. 3 + + /3.................. 0. +.......................................................... sin.............................. 3. sin cos 3....................... 4. 5. 8........................ + 3 arctan +.................. 6. sin arccos.................... 7. log {+ log + log }............ 8. +......................... 9. + +................... 0. log 3 + 3 +.............. 7 0 3
No.5 differentiation of to the e. a = a a. a = a log a 3. = 4. f g = log ep a > 0 a = e log a > 0 > 0 = e log 3 f, g f g f, g, f, g f > 0 4 logarithmic differentiation 5 a bc a bc a b c = a bc. e. 3. 4. log 5. log 6. log log 7. 8. 9. 0.. e............................... df d = f d log f d................................ 3............................... = log = log = log + f = {f { } p {f n} pn } df d = f f p f + + f n pn f n f = + 3 4 f = + 3 + 4 + 3 4 f = F u, v = u v, u =, v = f = F u, v. df d F u, v du = u d = uv u d d + uv d d = vu v + u v log u = + log = log + F u, v dv + d 4. e arctan........................... 5. arctan........................... 6. arctan........................... 7. sin.......................... 8. sin........................... 9. sin sin........................ 0. sin............................ sin............................ sin sin......................... 3. sin sin sin..................
No.6 = high-order differentials d df f = d d = f d d d d f = d f = f d n d d n d d f = dn f = f n n d n d n d n d f, f. n d,,.. +. log 3. arcsin 4. sinh.......................... 5. tan............................. 6. e.............................. of product of functions Leibniz { } n n n f g = f n i g i i n i i=0 = nc i = n! n i! i! : 0!=,!=,!= =, 3!=3 =6 : n n n =, = n, = 0 nn n, = nn n. 3 6 fg = f g + fg fg = f g + f g + fg fg = f g + 3f g + 3f g + fg fg = f g + 4f g + 6f g + 4f g + fg fg 5 = f 5 g + 5f 4 g + 0f 3 g + 0f g 3 + 5f g 4 + fg 5 n n +! 0 0, 0, 0. 3 n n.. e n n nth order differentials,. e n = e sin n = sin + nπ cos n = cos + nπ n n.. e n. 3. 4. 5. 6. e n n 3 sin + e n.................. 3 + cos n................ + a n n.................... e n 3. sin n 4. n 5. n..................................... n!!. n!! n double factorial factorial n n!! = nn n 4, n n!! = nn n 4., 0!! =!! =. n!! = n n!! n =,. 6. sin n.................................. sin = cos. k k = 0 e n = n 0 0 e n + n e n n = 0. 0 + 90e. 3 n = 0. 3 70 sin + 30 70 cos. 3. sin n = cos n = sin n = cos n 3 cos n = sin n = cos n = sin n 3, sin n = sin + nπ cos n = cos + nπ.
No.7 Taylor epansin f = fa + f a! a + f a! a + + f n a n! a n + R n+, R n+ = f n+ ξ n +! an+ ξ a. lim n R n+ = 0 f = n=0 f n a a n n! a = 0. R n+ O n+ o n. O, o, order. [] e = + +! + 3 3! + 4 4! + 5 5! + 6 6! + [] sin = 3 3! [3] cos =! [4] log + = [5] + s = s k + 4 4! + 5 5! 6 6! + + 3 3 4 4 + 5 5 6 6 + s k k, s 0 =, k=0 = ss s s k + [] [3], [4],[5] <. k! arctan 3. f = arctan f0 = 0 f = + f 0 = f = + f 0 = 0 f = 3 + 3 f 0 = f = f0 + f 0 + f 0! + f 0 3 + O 4 3! arctan = 3 3 + O 4. arctan,. 4 O 5. 3 No.8 arctan = 3 3 + 5 5 7 7 + 9 9 +. =, arctan = π 4, π = 4 3 + 5 7 + 9 + 3 5 + 7... 5 e i = cos + i sin. i, i =., Euler.,,,,,.,.,,., e i+y = e i e iy, cos + y + i sin + y = cos +i sin cos y+i sin y = cos cos y sin sin y+ isin cos y+cos sin y,, cos + y = cos cos y sin sin y, sin + y = sin cos y + cos sin y, sin cos. 6 0 = 0 4., 4 f = 0,. O 4. f = sin + f = e cos 3 f = e................... 4 f = + e................. 6 + /3 7 +......................... 8 +......................... 9.......................... + [5]. 0 4 + 3/ 4 3/ + 4 3/ [5] s 3, 4.
No.8 cosh = e + e. e = + + + 3 6 + 4 4 + 5 0 + 6 70 + e = + 3 6 + 4 4 5 0 + 6 70 + cosh = + + 4 4 + 6 70 +.,,. arctanh = + log log ± 6. = + + 4 6 + + 4n 4n+ + [0, ] arctan = 3 3 + 5. 5 7 7 + + 4n+ 4n+ 4n+3 4n+3 + 5 + log +. 6 / arcsin. 7 = 0. 4. log + 3. log + = 8 + O 3 + 3 3 + O 4 = + + 3 + 4 8 3 + O 4 = + 4 3 + O 4 e sin 3. e sin. e sin = + 3 + 6 O5 + 3 + 6 O5 + 6 + O 3 3 + 4 + O 3 4 + O 5., 3 + 6 O5 = 4 + 3 O5, + O 3 3 = 3 + O 5, + O 3 4 = 4 + O 6 e sin = + + 8 4 + O 5. 3 e cos. + cos +! cos + cos = + sin sin = 3 3! + 5 5! 7 4n+ 7! + + 4n+! 4n+3 4n+3! + cos cos =! + 4 4! 6 4n 6! + + 4n! 4n+ 4n+! +. 4 e k=0 k k!,. e = e. cos. sin 3. 3 + /3 4. + + 3 /3 5. cos 6. log + +................... log + α = α α + 3 α3 4 α4 + Oα 5 α = +. 7. log cos sin.................... 8. 9. cossin = + 5 4 4 + O 6. log + α α = + 5 4 4 +O 6. e +............................. e +. e sin........................ e sin. + α, + α. 0. arcsinh.......................... arcsinh = 0 dt +t., + t /.. tan............................. sin / cos, arctan.. e e...............................
No.9 limits of indeterminate forms [] 0 [] 0, 0, [3] 0 0,, 0 log [3] [], [] []. L Hospital s rule a f, g 0 ± f lim a g = lim f a g. a ±. i 0. ii,, limf g fg /g. Hospital Johann Bernoulli. WEB log lim.. partial differentiation f, y f, y : y. f. : y. f y. = f, y = f, y f, y = f, y f, y = f, y f, y = f, y f. f y. f y. f yy. f y = f y. f, y = 4 + 3 y + y, f = 4 3 + 6y + y, f y = 3 + 4y, f = 43 + 6y + y = + 6y f y = 43 + 6y + y = 6 + 4y f y = 3 + 4y = 6 + 4y f yy = 3 + 4y = 4 7 f, y = sin y + f, f y. t = y + f = d dt sin t t. lim log. +0 0. 3 lim cos. sin +0 0, cos sin 0. 4 lim +.., A > 0 A = eplog A. 5 y = > 0. +0., y. 8 f, y = y f, f y, f, f y, f y, f yy. a = a a, a = a log a. = a +, y = b + y, f, y = f a,b+f a,b +f y a,b y f a,b + f y a,b y + f yy a,b y +R 3 +! f, y, f = f y = { 0, f f yy f y > 0. > 0 f < 0. f f yy f y < 0. : stationary point, : saddle point f, y,. 6 y = / > 0. +0.,, y.. 9 f, y = + y + y +, y. 0 f, y = 3 y, y.
No.0 total differential formula = t, y = yt, z = z, y dz dt = z d dt + z dy dt. dz = d z t, yt, d = d t, dy = d yt dt dt dt dt dt dt z = z z, y, = z, y., dt z : dz = z z d + dy = u, v, y = yu, v z z u = u u z z z = z u, v, yu, v, z = z, y., z. u u u =,,... r [cm], m [g], ρ [g/cm 3 ] ρ = 3m. 4πr 3, r 0 cm, cm/s r=0, dr =., m 50 g, 8 g/s dt m=50, dm =8. dt ρ [g/cm 3 s] dρ. dt, π 3.4.,. t = 0 r = 0 + t + Ot, m = 50 + 8t + Ot. ρ Taylor t,,. R V W = V. R R, V, W W, W R, V, R, V., V/V = 0.0, R/R = 0.05, W/W. V + V W = R+ R V, R, R, V., V, R W/W. 4 = e u cos v, y = e u sin v. i T= u. ii, u, v, u u, v u,. iii Laplacian = + u, v, u,. 5 u = log + y, v = arctan y. i T = u u, y. ii u,, y,,. iii u +, y,,. 6 4 5,., T T.,,. 3 r, h V = πr h. r r, h h. r=.00 cm, h=.00 cm, r 0.05 cm, h 0.05 cm, V cm 3., V V/V, r h. πr + r h + h πr h,, r h. V 5 %, 9% r, 6 % h. 7 = r cos θ, y = r sin θ r = + y θ = arctan y, 4 6. 8 = u v, y = uv, + u, v, u,.
No. No.3 / 3 π 3 No.3 / 4 π 3 No.3 / 5 π 4 No.3 / 6 3 4 π No.3 / 7 π 3 No.3 / 8 π 3 No.3 / 9 3 5 5 No.3 / 0 3 No.3 / 3 No.3 / = 4 5 No.4 / 8 9 3 3 No.4 / 9 + 33++ /3 No.4 / 0 + No.4 / 3 No.4 / cos No.4 / 3 cos cos 3 3 sin sin 3 No.4 / 4 8 3 + 3 9 No.4 / 5 arctan + + No.4 / 6 cos arccos No.4 / 7 +log {+log+log } No.4 / 8 + 8 0. No.4 / 9 + +4 + 8 + + + No.4 / 0 3 3 + No.5 / 5- e No.5 / 5- log No.5 / 5-3 No.5 / 5-4 + e arctan No.5 / 5-5 log + arctan log + No.5 / 5-6 arctan log + arctan + No.5 / 5-7 cos No.5 / 5-8 log + cos No.5 / 5-9 sin cos log + sin cos sin No.5 / 5-0 sin cos No.5 / 5- sin logsin + cot No.5 / 5- sin sin cos logsin + No.5 / 5-3 sin sin sin +sin cos logsin logsin + + cot No.6 / / 4 cosh, cosh + 4 sinh, 8 3 cosh + sinh No.6 / / 5 cos, sin cos 3, 6 4 cos cos 4 No.6 / / 6 e, e, 4 3e. No.6 / / 5 n n 3!! n No.6 / / 6 n cos + nπ { No.6 / 3 / 4 n e + + n + + nn 4 No.6 / 3 / 5 { 3n 3n } cos + nπ +n{3 n 3n + } sin + nπ { No.6 / 3 / 6 n! + n + a + nn + a} No.7 / 3 + + 5 + 8 3 3 + O 4 No.7 / 4 + 4 + 3 3 + 7 384 3 + O 4 No.7 / 7 + 8 + 6 3 5 8 4 + O 5 No.7 / 8 + 3 8 5 6 3 + 35 8 4 + O 5 No.7 / 9 + + + 3 + 4 + O 5. No.8 / 7 / 6 + 3 3 + 4 4 + 5 5 3 6 + 7 7 + No.8 / 7 / 7 + 4 + 90 6 + 50 8 + No.8 / 7 / 8 + 5 8 3 3 + 65 4 4 63 60 5 + No.8 / 7 / 9 3 3 + 5 4 4 + 3 5 + 9 70 6 + No.8 / 7 / 0 6 3 + 3 40 5 + + n+ n n+ + = n=0 n n!! n!!n+ n+,, 0!! =!! =. No.8 / 7 / + 3 3 + 5 5 + 7 35 7 + 6 835 9 + No.8 / 7 / e + e + e + 5 6 e3 + 5 8 e4 + 3 30 e5 +. }