I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

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1 I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. :

2 I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π sin cos sin + cos = sin α ± β = sin α cos β ± cos α sin β cos α ± β = cos α cos β sin α sin β tan 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? =? 4 y = sin y = cos sinh cosh 5 π sin π cos 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 cosh sinh =, sinh α ± β =, cosh α ± β =

3 I No. log logarithmic function y = e < <, y > = log y y >, < < inverse trigonometric functions II = arcsin y y = sin π π, y = arccos y y = cos π, y = arctan y y = tan π < < π, < y < base a> y = a = log a y e log ln log log ln sin y Sin y sin y arcsin y arc Sin sin π 6 = arcsin = arccos = arctan π 4 = arcsin = arccos = arctan a, b, c,, y > 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 >, b >. 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. 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

4 I No. sin arccos y = sinh = arcsinh y 5 cos arctan y = cosh = arccosh y y = tanh = arctanh y tan arcsin 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 arcsin + arcsin sin arcsin 4 cos arccos 6 arcsin π arcsin π 5 arccos 6 arccos π 4 4 π 7 arctan π arctan π 9 sin arctan 4 cos arcsin tan arccos arccos 5 arccos = arccos. = differential:, differentiation:, derivative: f = df = lim f + h f h 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 dg f g = f g g 5. dy = f = dy f f 4 f/g = fhg, hy = y

5 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. tanh =. arcsinh = cosh, coth = sinh +. arccosh = ±. arctanh = 9 9 f = f g fg 8,9 g g 4 5 cos + sin = 9 cosh sinh = dy = dy differentiation of composite functions cos5 6. sin arcsin 4. arccos cos 5. arctansin 6. ep 5 7. log / / sin cos. sin cos.. cos cos sin sin arctan arctan sin arccos log + log + log cos arccos +log +log+log log

6 I No.5 differentiation of to the e. a = a a. a = a log a. = 4. f g = log ep a > a = e log a > > = e log f, g f g f, g, f, g f > 4 logarithmic differentiation df = f d log f = log = log = log + f = {f } p {f n} p n { } df = f f p f + + p f n n f n f = + 4 f = f = F u, v = u v, u =, v = f = F u, v. df F u, v du = u = uv u + uv = vu v + u v log u = + log = log + F u, v dv + 5 a bc a bc a b c = a bc. e.. 4. log 5. log 6. log log e e arctan 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. sin sin cos. sin.. sin logsin + cot. sin sin sin sin cos logsin +. sin sin sin... sin sin sin +sin cos logsin logsin + + cot

7 I No.6 high-order differentials d df f = = f d d f = d f = f n d d n f = dn f = f n n n d n d n f, f n,,. +. 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. of product of functions Leibniz n {fg} n n = f n i g i i i= n n! = i n C i = n i! i! :!=,!=,!= =,!= =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 +! 6 6 n n. e n. e n n sin + e n n + cos + a n n k k = e n = n e n + n e n 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 6 : n cos + nπ. sin = cos. n = + 9e n = 7 sin + 7 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}

8 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+ = f = n= f n a a n n! a = R n+ O n+ o n. f = f+f + f! + f +O 4! arctan = + O 4 arctan 4 O 5 No.8 arctan = = arctan = π 4 π = f = O 4 f = sin + f = e cos [] e = + +! +! + 4 4! + 5 5! + 6 6! + [] sin =! [] cos =! [4] log + = [5] + s = s k + 4 4! + 5 5! 6 6! s k k, s =, k= = ss s s k + arctan [] [] [4],[5] < f = arctan f = f = + f = 5 e i = cos + i sin f = i i = + f = Euler f = + f = k! 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 = / / [5] f = e f = O 4 4 f = + e f = O : O : O 5 + : O 5

9 I No.8 cosh = e + e e = e = cosh = arctanh = + log log ± 6 = n 4n+ + [, ] arctan = n+ 4n+ 4n+ 4n+ + 5 / arcsin Taylor. 6 = Taylor 4.. cos log + log + = 8 + O + + O 4 = O 4 = O 4 e sin. sin. + / / 5. cos 6. log log cos sin e sin e sin = O O O O 4 + O 5., + 6 O5 = 4 + O5, + O = + O 5, + O 4 = 4 + O 6 e sin = O 5 e cos + cos +! cos + cos = + sin Taylor sin =! + 5 5! 7 4n+ 7! + + 4n+! 4n+ 4n+! + cos Taylor cos =! + 4 4! 6 4n 6! + + 4n! 4n+ 4n+! +. 4 e Taylor. e = e e + e sin. arcsinh. tan. e e 6 : log + + = : cossin = O 6 logcossin = O 6 8 : e + = : e sin = : arcsinh = dt/ + t, arcsinh = n n+ n+ + = n= n n!! n!!n+ n+,,!! =!! =. : tan = sin / cos, arctan. : e e = e + e + e e e4 + e5 +

10 I No.9 limits of indeterminate forms [] [],, [],, log [] [], [] []. L Hospital s rule a f, g ± f lim a g = lim f a g. a ±. i. ii,, limf g fg /g. Internet Johann Bernoulli. Hospital, Bernoulli 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.. lim log. +. lim cos. sin +, cos sin. 4 lim +.., A > A = eplog A. 5 y = >. +., y. 6 y = / >. +.,, y.. 7 f, y = sin y + f, f y. t = y + f d = dt sin t t. 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 =, f f yy fy > f >. <. f f yy fy <, saddle point 9 f, y = + y + y +, y. f, y = y, y.

11 I No. total differential formula = t, y = yt, z = z, y dz dt = z dt + z dy dt. dz = d z t, yt, = d t, dy = d yt dt dt dt dt dt dt z = z z, y, = z, y., dt z : dz = z z + 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 cm cm/s r=, dr =. m 5 g 8 g/s dt m=5, dm =8. dt ρ [g/cm s] dρ. dt π.4.. R V W = V. R R, V, W W, W R, V, R, V., V/V =., R/R =.5, W/W. V + V W = R+ R V, R, R, V., V, R W/W. r, h V = πr h. r r, h h. r=. cm, h=. cm, r.5 cm, h.5 cm, V cm., V V/V, r h. πr + r h + h πr h,, r h. V 5 %, 9% r, 6 % h. 4 = e u cos v, y = e u sin v. i T=. u u u, v ii, u, v, u,. iii + u, v, u,. 5 u = log + y, v = arctan y. i T =. u u, y ii u,, y,,. iii u +, y,, ,., T T.,,. 7 = r cos θ, y = r sin θ r = + y. 8 = u v, u, v, θ = arctan y, 4 6 u,. y = uv, +

5 36 5................................................... 36 5................................................... 36 5.3..............................

5 36 5................................................... 36 5................................................... 36 5.3.............................. 9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................

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