I 0 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No.0 :
I No. sin cos sine, cosine : trigonometric function π : π =.4 : n = 0, ±, ±, sin + nπ = sin cos + nπ = cos : parity sin = sin : odd cos = cos : even. sin. sin. sin + π sin cos Pythagoras sin + cos = addition formulas sin α ± β = sin α cos β ± cos α sin β cos α ± β = cos α cos β sin α sin β cotangent : cot = cos sin = tan secant : sec = cos cosecant : cosec = sin co 7 tan cot cosec sec e : eponential function = 8 eponent y = e : = ep : e =.78 : 8 y = ep 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 tan? = 0? 4 y = sin y = cos sinh cosh 5 π 0 sin d π 0 cos d cosh sinh = 6 sin = 4 sin + 4 sin sin 4 = 8 cos 4 cos + 8 cos, cos 4 tangent : tan = sin cos 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 =
I 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 = arccos = arctan π 4 = arcsin = arccos = arctan π = arcsin = arccos = arctan a, b, c,, y > 0 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.. a > 0, b > 0. log a. log a a. 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, e a = e log e a log a = log e log e a,,.,,... arcsin. arcsin. arccos 4. arccos 5. arctan 6. arctan 7. sin arccos 8. cos arcsin
I No. sin arccos y = sinh = arcsinh y 5 cos arctan y = cosh = arccosh y y = tanh = arctanh y tan arcsin 0 log : 4 arcsin = arctan arcsinh = log + +, < < 5 arccos = arcsin arccosh = log ±, 4 6 = ± log + 6 arctan = arccos 6 7 cos arcsin arcsin arctanh = log +, < < 4 8 arcsin + arcsin arcsinh arccosh arctanh = arcsin [ π, π ] 9 arctan + arctan 0 arcsin + arcsin sin arcsin 4 cos arccos 6 arcsin π...................... 4 arcsin................ π 5 arccos 6 arccos....................................... π 4 4 π 7 arctan π...................... 8 arctan................ π 9 sin arctan 4 0 cos arcsin............ tan arccos arccos 5 arccos 4 5............ 5 5......... = arccos. = 4 5. 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 f 4 = + + +. 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 4 f/g = fhg, hy = y
I No.4 derivatives of basic functions. s = s s. e = e. 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 = ±. arctanh = 9 9 f = f g fg, g g 4 5 cos + sin = 9 0 cosh sinh = dy d = d dy 4 4 6 7 5 5 8 6 9 7 0 7 differentiation of composite functions 8.. 4 cos 5 6. sin arcsin 4. arccos cos 5. arctan sin { 6. ep } 5 7. log 4 + + + 5 8...... 9 9. + + /.......... 0. + + ++ /........................ +............................. sin........................ cos. sin cos.. cos cos sin sin 4. 5. 8 8................ + + 9 arctan +.... arctan + + 6. sin arccos............... 7. log + log + log. 8. 9. cos arccos +log +log+log +............... + +. 8 + + +4 + 0. log + +.......... + + + + 8 0
I No.5 differentiation of to the e. a = a a. a = a log a. = 4. f g = log ep a > 0 a = e log a > 0 > 0 = e log f, g f g f, g, f, g f > 0 4 logarithmic differentiation df d = f d log f d = log = log = log + f = {f } p {f n} p n { } df d = f f p f + + p f n n f n f = + 4 f = + + 4 + 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 5 a bc a bc a b c = a bc. e.. 4. log 5. log 6. log log 7. 8. 9. 0.. e................................................................ 4. e arctan.................. 5. arctan.................. e log log + + e arctan log + arctan 6. arctan..... arctan log + arctan + 7. sin................... cos 8. sin......... log + cos 9. sin sin................................ sin cos log + sin cos sin 0. sin................... sin cos. sin.. sin logsin + cot. sin sin..................................... sin sin cos logsin +. sin sin sin... sin sin sin +sin cos logsin logsin + + cot
I 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. arcsin 4. sinh 5. tan 6. e 4 : cosh, cosh + 4 sinh, 5 : 8 cosh + sinh. cos, sin cos, 6 4 cos cos 4 6 : e, e, 4 e. n nth order differentials e n = e sin n = sin + nπ cos n = cos + nπ n n. e n. e n.. sin n 4. n 5. n 6. sin n 5 : n n!! n. n!! n double factorial, factorial n n!! = nn n 4 n n!! = nn n 4 0!! =!! = n!! = n n!! n =, 6 : n cos + nπ. sin = cos. of product of functions Leibniz { } n n n f g = f n i g i i n i i=0 = n C i = n! n i! i! : 0!=,!=,!= =,!= =6 : n n n =, = n, = 0 nn n, = nn n. 6 fg = f g + fg fg = f g + f g + fg fg = f g + f g + f g + fg fg = f g + 4f g + 6f g + 4f g + fg n n +! 0 0, 0, 0. n n. e n.. 4. 5. 6. e n n sin + e n n + cos + a n n k k = 0 e n = n 0 0 e n + n e n n = 0 0 + 90e n = 0 70 sin + 0 70 cos sin n = cos n = sin n = cos n cos n = sin n = cos n = sin n sin + nπ cos + nπ { } 4 : n e + + n + + nn 4 5 : { n n } cos + nπ +n{ n n + } sin + nπ { 6 : n! + n + a + nn + a}
I 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 = + +! +! + 4 4! + 5 5! + 6 6! + [] sin =! [] cos =! [4] log + = [5] + s = s k + 4 4! + 5 5! 6 6! + + 4 4 + 5 5 6 6 + s k k, s 0 =, k=0 = ss s s k + [] [], [4],[5] <. k! arctan. f = arctan f0 = 0 f = + f 0 = f = + f 0 = 0 f = + f 0 = f = f0+f 0+ f 0! + f 0 +O 4! arctan = + O 4. arctan,. 4 O 5. No.8 arctan = + 5 5 7 7 + 9 9 +. =, arctan = π 4, π = 4 + 5 7 + 9 + 5 + 7..., 4 f = 0,. O 4. f = sin + f = e cos f = e f = + + 5 + 8 + O 4 4 f = + e f = + 4 + + 7 84 + O 4 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. 6 + / 7 +, : + 8 + 6 5 8 4 + O 5 8 +, : + 8 5 6 + 5 8 4 + O 5 9 + [5]. : + + + + 4 + O 5 0 4 + / 4 / + 4 / [5] s, 4.
I No.8 cosh = e + e e = + + + 6 + 4 4 + 5 0 + 6 70 + e = + 6 + 4 4 5 0 + 6 70 + cosh = + + 4 4 + 6 70 + arctanh = + log = + + 4 6 + + 4n 4n+ + [0, ] arctan = + 5 5 7 7 + + 4n+ 4n+ 4n+ 4n+ + 5 + log +. 6 / arcsin. 7 = 0 log ± 6 4. log + log + = 8 + O + + O 4 = + + + 4 8 + O 4 = + 4 + O 4 e sin. cos. sin. + / 4. + + / 5. cos 6. log + + e sin e sin = + + 6 O5 + + 6 O5 + 6 + O + 4 + O 4 + O 5., + 6 O5 = 4 + O5, + O = + O 5, + O 4 = 4 + O 6 e sin = + + 8 4 + O 5 e cos + cos +! cos + cos = + sin sin =! + 5 5! 7 4n+ 7! + + 4n+! 4n+ 4n+! + cos cos =! + 4 4! 6 4n 6! + + 4n! 4n+ 4n+! +. 4 e. e = e 7. log cos sin 8. 9. e + e sin 0. arcsinh. tan. e e 6 : log + + = + + 4 4 + 5 5 + 7 : cossin = + 5 4 4 + O 6 logcossin = + 4 + O 6 8 : e + = + 5 8 + 65 4 4 6 60 5 + 9 : e sin = + 5 4 4 + 5 + 9 70 6 + 0 : arcsinh = dt/ + t, 0 arcsinh = 6 + 40 5 + + n n+ n+ + = n=0 n n!! n!!n+ n+,, 0!! =!! =. : tan = + + 5 5 + 7 5 7 + 6 85 9 + sin / cos, arctan. : e e = e + e + e + 5 6 e + 5 8 e4 + 0 e5 +
I No.9 limits of indeterminate forms [] 0 [] 0, 0, [] 0 0,, 0 log [] [], [] []. 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 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 + y + y, f = 4 + 6y + y, f y = + 4y, f = 4 + 6y + y = + 6y f y = 4 + 6y + y = 6 + 4y f y = + 4y = 6 + 4y f yy = + 4y = 4 lim log.. 7 f, y = sin y + f, f y. t = y + f d = dt sin t t. lim log. +0 0. lim cos. sin +0 0, cos sin 0. 4 lim +.., A > 0 A = eplog A. 5 y = > 0. +0., y. 6 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 +! 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 9 f, y = + y + y +, y. 0 f, y = y, y.
I 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 ] ρ = m. 4πr, r 0 cm, cm/s r=0, dr =., m 50 g, 8 g/s dt m=50, dm =8. dt ρ [g/cm s] dρ. dt, π.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=. ii u, 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.,,. 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., 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,.