3 3.1 3.1.1 A f(a + h) f(a) f(x) lim f(x) x = a h 0 h f(x) x = a f 0 (a) f 0 (a) = lim h!0 f(a + h) f(a) h = lim x!a f(x) f(a) x a a + h = x h = x a h 0 x a 3.1 f(x) = x x = 3 f 0 (3) f (3) = lim h 0 ( = lim h 0 3 + h 3 h 3 + h 3)( 3 + h + 3) h( 3 + h + 3) (3 + h) 3 = lim h 0 h( 3 + h + 3) 1 = lim = 1 h 0 3 + h + 3 2 3 f(x) = x x = 0 79
80 3 3.1 f(x) = x (1) f (1) (2) f (2) f(x) x = a y y = f(x) f (a) y = f(x) A(a, f(a)) f(a + h) 1 3.2 f(x) = A f(a) A x (3, 3) O a a + h x 1 f(x) x = a y = f(x) A(a, f(a)) x
3.1. 81 B f(x) f(x) x = a x = a x a f(x) f(a) = f(x) f(a) (x a) x a f(x) x = a f(x) f(a) lim x a x a lim(x a) = 0 x a = f (a) lim x a {f(x) f(a)} = f (a) 0 = 0 lim f(x) = f(a) x a f(x) x = a f(x) x = a x = a 3.2 f(x) = x lim f(x) = f(0) x 0 f(x) x = 0 f(x) = x f(0 + h) f(0) h = h h h lim h +0 lim h 0 h = lim h +0 h h = lim h 0 h h = 1 h h = 1 y y = x 1 1 O 1 x f(0 + h) f(0) lim f (0) h 0 h f(x) = x x = 0
82 3 3.3 f(x) x = 1 (1) f(x) = x 1 (2) f(x) = x 2 1 C f(x) x f(x) f(x) a f (a) f(x) f 0 (x) f(x) f (x) f(x) f 0 f(x + h) f(x) (x) = lim h!0 h y = f(x) y 0 dx d f(x) dx y = f(x) x h x x x y f(x + x) f(x) y y y f 0 (x) = lim x!0 x
3.1. 83 3.3 f(x) = 1 x ( f 1 1 (x) = lim h 0 h x + h 1 ) x { } 1 x (x + h) = lim h 0 h (x + h)x 1 = lim h 0 (x + h)x = 1 x 2 3.4 (1) f(x) = 1 2x (2) f(x) = x
84 3 3.1.2 A f(x) f (x) f(x) g(x) 1 {k f(x)} 0 = k f 0 (x) k 2 {f(x) + g(x)} 0 = f 0 (x) + g 0 (x) 3 {f(x) g(x)} 0 = f 0 (x) g 0 (x) 4 {f(x)g(x)} 0 = f 0 (x)g(x) + f(x)g 0 (x) 1 3 II 4 4 {f(x)g(x)} = lim h 0 f(x + h)g(x + h) f(x)g(x) h {f(x + h) f(x)}g(x + h) + f(x){g(x + h) g(x)} = lim h 0 h { } f(x + h) f(x) + h) g(x) = lim g(x + h) + f(x) g(x h 0 h h f(x) g(x) f(x + h) f(x) lim h 0 h = f (x), lim h 0 g(x + h) g(x) h = g (x) lim g(x + h) = g(x) h 0 {f(x)g(x)} = f (x)g(x) + f(x)g (x)
3.1. 85 x n II (x) = 1, (x 2 ) = 2x, (x 3 ) = 3x 2 c (c) = 0 4 x 4 3.4 (x 4 ) = (x 3 x) = (x 3 ) x + x 3 (x) = 3x 2 x + x 3 1 (x 4 ) = 4x 3 3.5 3.4 (1) (x 5 ) = 5x 4 (2) (x 6 ) = 6x 5 2 x n n (x n ) 0 = nx n`1 2 n = k (x k ) = kx k 1 (x k+1 ) = (x k x) = (x k ) x + x k (x) = kx k 1 x + x k 1 = (k + 1)x k n = k + 1
86 3 3.1 (1) y = 2x 5 5x 4 (2) y = (x 2 3)(4x 2 + 5) (1) y = 2 5x 4 5 4x 3 = 10x 4 20x 3 (2) y = (x 2 3) (4x 3 + 5) + (x 2 3)(4x 2 + 5) 4 = 2x(4x 2 + 5) + (x 2 3) 8x = 8x 3 + 10x + 8x 3 24x = 16x 3 14x 3.6 (1) y = x 5 + 2x 4 (2) y = 3x 6 4x 3 (3) y = (x + 1)(x 3 4x) (4) y = (3x 2 2)(x 2 + x + 1)
3.1. 87 B x x 84 4 f(x) g(x) { } 1 0 5 = g0 (x) g(x) {g(x)} 2 { } f(x) 0 6 = f 0 (x)g(x) f(x)g 0 (x) g(x) {g(x)} 2 1 5 g(x) 0 g(x) g(x) = 1 4 { } g 1 1 (x) g(x) + g(x) = 0 g(x) { } 1 g(x) = g (x) g(x) g(x) { } 1 = g (x) g(x) {g(x)} 2 3.7 f(x) g(x) = f(x) 1 g(x) 4 5 6
88 3 3.2 (1) y = 1 3x + 2 (1) y (3x + 2) = (3x + 2) = 3 2 (3x + 2) 2 (2) y = (x2 ) (x 1) x 2 (x 1) (x 1) 2 = = x2 2x (x 1) 2 (2) y = x2 x 1 2x(x 1) x2 1 (x 1) 2 3.8 (1) y = 1 2x 3 (2) y = x2 x + 3 (3) y = 2x 1 x 2 + 1
3.1. 89 (x n ) = nx n 1 n n n = m m ( ) 1 (x n ) = = (xm ) mxm 1 x m (x m ) = xm 1 2 x 2m x 2m = x(m 1) 2m = x m 1 = mx m 1 = nx n 1 x n n (x n ) 0 = nx n`1 n = 0 x 0 = 1 3.5 ( ) 1 = (x 2 ) = 2x 2 1 = 2x 3 = 2 x 2 x 3 3.9 (1) y = 3 x (2) y = 2 x 3 (3) y = 4 x 2
90 3 C y = (x 3 + 1) 2 u = x 3 + 1 y = u 2 y u y 2 y = u 2, u = x 3 + 1 y = f(g(x)) 2 y = f(u) u = g(x) u y = f(u) x u = g(x) x x u u u u y y y x y x = u u y x u = g(x) x 0 u 0 dx = lim y ( x 0 x = lim u u y ) x 0 x y = lim u 0 u lim u x 0 x = du du dx y = f(u) u u = g(x) x y = f(g(x)) dx = du du dx
3.1. 91 3.3 y = (3x 2 2) 5 u = 3x 2 2 y = u 5 du = 5u4, du dx = 6x dx = du du dx = 5u4 6x = 30x(3x 2 2) 4 3.10 (1) y = (3x + 1) 4 (2) y = (1 2x 2 ) 3 (3) y = 1 (4x + 3) 2
92 3 dx = {f(g(x))}, du = f (u) = f (g(x)), du dx = g (x) {f(g(x))} 0 = f 0 (g(x))g 0 (x) 3.3 y = (3x 2 2) 5 y = 5(3x 2 2) 4 (3x 2 2) = 5(3x 2 2) 4 6x = 30x(3x 2 2) 4 3.11 a b (1) y = (ax + b) 6 (2) y = 1 (ax + b) 3
3.1. 93 3.12 a b n (1) d dx f(ax + b) = af (ax + b) (2) d dx {f(x)}n = n{f(x)} n 1 f (x) D 3.6 y = 4 x x y = x 4 (x = 0) x = y 4 1 y = 4 x d dx y4 = d y4 dx = 4y3 dx 1 x 1 = 4y 3 dx y = 4 x dx = 1 4y = 1 3 4 4 x 3
94 3 f(x) g(x) y = f(x) x x = g(y) x 1 = d dx g(y) 1 = d g(y) dx g(y) = x 1 = dx dx dx = 1 dx 3.6 y = 4 x y = 4 x x x = y 4 dx = 1 = dx 1 4y 3 = 1 4( 4 x) 3 = 1 4 4 x 3 3.13 (1) y = 6 x (2) y = 3 x (x > 0)
3.1. 95 E x p y = 4 x y = x 1 4 n y = x 1 n y = x 1 n x x = y n dx = 1 dx = 1 ny n 1 1 y = 1 n 1 ( ) n 1 = 1 x 1 n dx = 1 n x 1 n 1 x 1 1 n = x 1 n 1 ( ) 1 x 1 n = n x 1 n 1 (x n ) = nx n 1 n 1 n
96 3 x p p (x p ) 0 = px p`1 p p = m n ( ) n m m x p = x m n = x 1 n ( ) 1 x 1 n = n x 1 n 1 (x p ) = {( ) m } x 1 n ( ) m 1 ( ) = m x 1 n x 1 n ( ) m 1 1 = m x 1 n n x 1 n 1 = m n x m n 1 = px p 1 «m 1 1 + n n 1 m = n 1 ««1 + n n 1 = m n 1 3.7 y = 4 x 3 y = 4 x 3 y = x 3 4 y = 3 4 x 3 4 1 = 3 4 x 1 4 = 3 4 4 x 3.14 (1) y = x (2) y = 3 x 2 (3) y = 1 x
3.1. 97 3.1.3 1 n y = x n (a + b) n = n C 0 a n + n C 1 a n 1 b + + n C k a n k b k + + n C n b n 2 y = f(x)g(x)h(x) y = f (x)g(x)h(x) + f(x)g (x)h(x) + f(x)g(x)h (x) y = (x 2 + 1)(x + 2)(3x 4)
98 3 3 ( (1) y = x 1 ) 3 x (2) y = 4 x 2 (3) y = 1 1 x 2 [ (x + h) n x n 1 h = n C 1 x n 1 + n C 2 x n 2 h + + n C n 1 xh n 2 + h n 1 ] 2 ( ) y = 12x 3 + 6x 2 10x + 2 ( 3 (1) y = 3 x 1 ) 2 (1 + 1x ) (2) y x = x 2 4 x 2 (3) y = x (1 x 2 ) 1 x 2
3.2. 99 3.2 3.2.1 A sin x (sin x) = lim h 0 sin(x + h) sin x h sin(x + h) sin x = sin x cos h + cos x sin h sin x = (cos h 1) sin x + cos x sin h ( cos h 1 (sin x) = lim sin x + sin h ) h 0 h h cos x cos h 1 lim h 0 h = 0 lim h 0 sin h h = 1 3 (sin x) = 0 sin x + 1 cos x = cos x 3.15 cos x (cos x) = sin x tan x tan x = sin x cos x ( ) sin x (tan x) = = (sin x) cos x sin x (cos x) cos x cos 2 x 3 lim h 0 cos h 1 h = cos2 x + sin 2 x cos 2 x = 1 cos 2 x sin h = 0 63 2.7 lim = 1 61 h 0 h
100 3 (sin x) 0 = cos x (cos x) 0 = sin x (tan x) 0 = 1 cos 2 x 3.4 (1) y = sin 3x (2) y = cos 2 x (3) y = 1 tan x (1) y = cos 3x (3x) = 3 cos 3x (2) y = 2 cos x (cos x) = 2 cos x ( sin x) 2 sin x cos x = sin 2x = sin 2x (3) y (tan x) = tan 2 x = 1 tan x 2 = 1 sin 2 x 1 cos 2 x tan 2 x = sin2 x cos 2 x (1) f(x) = sin x y = f(3x) f (x) = cos x y = f (3x) (3x) = 3 cos 3x (2), (3) 3.16 (1) y = cos 2x (2) y = ( 2 sin 3x + π ) 4 (3) y = sin 2 x
3.2. 101 (4) y = tan 2 x (5) y = 1 sin x (6) y = cos 2 3x 3.17 (1) y = x sin x + cos x (2) y = x cos x sin x
102 3 B a 1 log a x (log a x) log = lim a (x + h) log a x h 0 { ( h 1 = lim h 0 h log a 1 + h )} x { ( 1 = lim h 0 x x h log a 1 + h )} x h x = k h 0 k 0 (log a x) = 1 x lim k 0 log a(1 + k) 1 k 1 k 0 (1 + k) 1 k k (1 + k) 1 k k (1 + k) 1 k 0.1 2.593742 0.1 2.867971 0.01 2.704813 0.01 2.731999 0.001 2.716923 0.001 2.719642 0.0001 2.718145 0.0001 2.718417 0.00001 2.718268 0.00001 2.718295 k 0 (1 + k) 1 k e e = lim(1 + k) 1 k e = lim 1 + 1 «n k 0 n n e e = 2.71828182845 e 1 (log a x) = 1 x log a e = 1 x log e a e (log e x) = 1 x log e e = 1 x log e e = 1
3.2. 103 e log e x e log x 1 (log x) 0 = 1 x 2 (log a x) 0 = 1 x log a 3.5 (1) y = log(2x + 3) (2) y = x log 2 x (1) y = 1 2x + 3 (2x + 3) = 2 2x + 3 (2) y = (x) log 2 x + x(log 2 x) 1 = log 2 x + x x log 2 = log 2 x + 1 log 2 (log f(x)) = f (x) f(x) 3.18 (1) y = log 3x (2) y = log 2 (2x 1) (3) y = log(x 2 + 1) (4) y = x log x x
104 3 log x x > 0 log x = log x (log x ) = (log x) = 1 x x < 0 log x = log( x) (log x ) = {log( x)} = 1 x ( x) = 1 x x log x (log x ) = 1 x 3.19 (log a x ) = 1 x log a 3 (log x ) 0 = 1 x 4 (log a x ) 0 = 1 x log a 3.6 (1) y = log cos x (2) y = log 2 x 2 1 (1) y = 1 cos x (cos x) = sin x cos x (2) y = 1 (x 2 1) log 2 (x2 1) = = tan x 2x (x 2 1) log 2
(log f(x) ) = f (x) f(x) 3.20 (1) y = log 2x + 3 (2) y = log sin x 3.2. 105 (3) y = log 4 2x 1 (4) y = log 2 x 2 4 3.1 α (x α ) = αx α 1 x > 0 x α > 0 y = x α log y = α log x x x α > 0 y = x α log y = α log x x y y = α 1 x y = α y x (x α ) = α xα x = αxα 1 (log y) = y y
106 3 3.21 3.1 (1) y = x x (x > 0) (2) y = x2 + 1 x C a 1 y = a x a x > 0 y = a x log y = x log a x y y = log a y = y log a (a x ) = a x log a a = e log a = log e = 1 (e x ) = e x 1 (e x ) 0 = e x 2 (a x ) 0 = a x log a
3.7 (1) y = e 3x (2) y = x 2 x (1) y = e 3x (3x) = 3e 3x (2) y = (x) 2 x + x(2 x ) = 1 2 x + x 2 x log 2 = 2 x (1 + x log 2) 3.2. 107 3.22 (6) a 1 (1) y = e 2x (2) y = e x2 (3) y = 3 x (4) y = 2 3x (5) y = xe x (6) y = (2x 1)a x
108 3 3.2.2 n y = f(x) f (x) x f (x) y = f(x) 2 y 00 f 00 (x) d2 y dx 2 d2 dx 2 f(x) f (x) y = f(x) 3 y 000 f 000 (x) d3 y dx 3 d3 dx 3 f(x) y f (x) 1 3.8 (1) y = sin x y = cos x y = sin x y = cos x (2) y = e x y = e x y = e x y = e x 3.23 3 (1) a 0 (1) y = ax 3 (2) y = 1 x (3) y = cos x (4) y = log x
(5) y = e x (6) y = e 2x 3.2. 109 y = f(x) n y = f(x) n y (n) f (n) (x) dn y dx n dx dn n f(x) y (1) y (2) y (3) y y y y = e x y (n) = ( 1) n e x 3.24 n (1) y = x n (2) y = e 2x
110 3 3.2.3 A x y y 2 = 4x y y = 2 x y y = ±2 x O x 2 y = 2 x 1 y = 2 x y = 2 x 2 1 dx = 2 1 2 x = 1 x = 2 y 2 ( x) = x 1 1 2 = 2 x 2 1 1 = 1 2 x 2 1 = 1 2 x dx = 2 1 2 x = 1 = 2 x y y 2 = 4x dx = 2 y y 0
3.2. 111 3.25 x 2 + y 2 = 1 (1) y (2) dx = x y
112 3 x y x y dx x 2 3.2 9 + y2 4 = 1 x y dx y 0 dx = 4x 9y y x x x 2 d dx y2 = d y2 = 2y dx dx 9 + y2 4 = 1 x 2x 9 + 2y 4 dx = 0 y 0 3 y 2 O x 2 9 + y2 4 = 1 3 x dx = 4x 9y 2 3.2 x 2 + y 2 = 9 x y 2 3 3.26 x y dx (1) y 2 = x (2) x 2 y 2 = 1
3.2. 113 3.27 a b a + y2 2 b = 1 x y 2 dx = b2 x a 2 y x 2 B C P(x, y) x y 1 t C 3.9 C P(x, y) x y t x = 2t 2 y = 2t 1 2 1 2 t y 2 = 4t 2 y 2 O 2 2 x y 2 = 2x y 2 = 2x C y 2 = 2x
114 3 C P(x, y) x y 1 t f(t) g(t) x = f(t), y = g(t) C t t 3.10 a y x 2 + y 2 = a 2 θ x = a cos θ, y = a sin θ a a O a θ (x, y) a x a θ C t x = f(t), y = g(t) y x dx = dt dt dx = dt 1 dx dt x = f(t) y = g(t) dx = dt dx dt = g 0 (t) f 0 (t)
3.2. 115 3.8 x y t t dx (1) x = 2t 2 y = 4t (2) x = cos t y = sin t dx (1) dt = 4t dt (2) dx dt = sin t dt = 4 dx = 4 4t = 1 t = cos t dx = cos t sin t = 1 tan t x y (1) dx = 4 y (2) dx = x y 3.28 x y t dx t (1) x = 2t y = t 2 1 (2) x = 3 cos t y = 2 sin t
116 3 3.2.4 4 (6) a 1 (1) y = 1 1 + cos x (2) y = sin 2 x cos 2x (3) y = (log x) 2
(4) y = log x + 1 x + 2 3.2. 117 (5) y = log e x e x + 1 (6) y = a 2x+1
118 3 5 a d dx log(x + x 2 + a) = 1 x2 + a 6 x u v 2 (uv) = u v + 2u v + uv 4 (1) y = (4) y = sin x (2) y = sin 2x(1 4 sin 2 x) (3) y = 2 log x (1 + cos x) 2 x 1 (x + 1)(x + 2) (5) y = e x (e x + 1) 2 (6) y = 2a 2x+1 log a 5 u = x + x 2 + a y = log u 6 (uv) = u v + uv (uv) = (u v + uv )
3.3. 119 3.3 3.3.1 A 1 y = x x 2 (1) y = x2 + x + 1 x (2) y = ( x + 1 ) 4 x
120 3 (3) y = 1 + cos x (4) y = sin x x (5) y = x e e x (x > 0) (6) y = 2 log x
3.3. 121 3 n x 1 1 + x + x 2 + + x n = 1 xn+1 1 x x x 1 1 + 2x + 3x 2 + + nx n 1 4 f(x) = sin x ( f (n) (x) = sin x + nπ ) 2
122 3 5 y = e x (sin x + cos x) y 2y + 2y = 0 x 2 6 a b a y2 2 b = 1 x y 2 dx = b2 x a 2 y
3.3. 123 3.3.2 B 7 f(x) f(a + h) f(a h) lim h 0 h = 2f (a) 8 (1) lim x 0 log(1 + x) x (2) lim x 0 e x 1 x 9 (1) y = 1 tan x 1 + tan x
124 3 (2) y = x 2 (log x) 3 (3) y = ex e x e x + e x (x + 2)(x + 3)3 f (x) log f(x) x 2 + 1 f(x) f (x) 10 f(x) =
11 a b t y = a cos 2t + b sin 2t d2 y dt 2 k 3.3. 125 = ky 12 x 2 3 + y 2 3 = 1 x ( y y dx = x ) 1 3 7 lim h 0 f(a h) f(a) h = f (a) 8 (1) log 1 = 0 (2) e 0 = 1
126 3 1 y = 3 x 2 [ y (x + h) x + h x x = lim h 0 h 2 (1) y = 3x2 + x 1 2x x (4) y = (x + h) 3 x 3 = lim h 0 h{(x + h) x + h + x x} ( (2) y = 4 x + 1 ) 3 (1 1x ) x 2 ] (3) y sin x = 2 1 + cos x x cos x sin x (5) y = (ex e 1 + x e )e x (6) y = 2log x log 2 x 2 x 3 nxn+1 (n + 1)x n + 1 (1 x) 2 [ ( 4 n = k f (k) (x) = sin x + kπ ) ( 2 f (k+1) (x) = cos x + kπ ) {( = sin x + kπ ) + π } ] 2 2 2 5 y = 2e x cos x y = 2e x (cos x sin x) [ 2x 6 x a 2y ] 2 b 2 dx = 0 [ ] f(a + h) f(a) f(a h) f(a) 7 lim + lim h 0 h h 0 h 8 (1) 1 (2) 1 (1) f(x) = log(1 + x) f (0) (2) g(x) = e x g (0) 9 (1) y 2 = (2) y = 2x(log x) 3 + 3x(log x) 2 (cos x + sin x) 2 (3) y = 4 (e x + e x ) 2 10 f (x) f(x) = 2x3 x 2 8x + 9 (x + 2)(x + 3)(x 2 + 1) f (x) = (x + 3)2 (2x 3 x 2 8x + 9) (x 2 + 1) 2 [ ] 11 k = 4 dt = 2a sin 2t + 2b cos y 2t d2 = 4a cos 2t 4b sin 2t dt2 [ 2 12 x 3 x 1 3 + 2 ] 3 y 1 3 dx = 0