3 }t. (1) (f + g) = f + g, (f g) = f g. (f g) = f g + fg, ( f g ) = f g fg g 2. (2) [f(g(x))] = f (g(x)) g (x) (3) d. = nv dx.

Μέγεθος: px

Εμφάνιση ξεκινά από τη σελίδα:

Download "3 }t. (1) (f + g) = f + g, (f g) = f g. (f g) = f g + fg, ( f g ) = f g fg g 2. (2) [f(g(x))] = f (g(x)) g (x) (3) d. = nv dx."

Καρπός Αλαφούζος
7 χρόνια πριν
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1 3 }t! t : () (f + g) f + g, (f g) f g (f g) f g + fg, ( f g ) f g fg g () [f(g(x))] f (g(x)) g (x) [f(g(h(x)))] f (g(h(x))) g (h(x)) h (x) (3) d vn n dv nv (4) dy dy, w v u x íªƒb N úb5} : () (e x ) e x () (ln x) x, x > 0 (3) (ln f(x) ) f (x) f(x) (4) (log a x) ( ln x ln a ) (ln a)x, x > 0 (5) (a x ) (e ln ax ) (e x ln a ) e x ln a ln a a x ln a, a > 0 (6) (f(x) g(x) ) (e g(x) ln f(x) ) f(x) g(x) (g (x) ln f(x) + g(x) f(x) f (x))

2 úiƒb5} : () (sin x) cos x () (cos x) sin x (3) (tan x) sec x (4) (cot x) csc x (5) (sec x) sec x tan x (6) (csc x) csc x cot x úiƒb5} : () (sin x) x, x < () (cos x) x, x < (3) (tan x) + x, x R (4) (cot x) + x, x R (5) (sec x) (6) (csc x) x x, x > x x, x > Â(ƒb : () sin hx ex e x (3) tan hx ex e x e x + e x () cos hx ex + e x (4) cot hx ex + e x e x e x (x 0) (5) sec hx e x + e x (6) csc hx e x e x (x 0)

x) + x, x R (5) (sec x) (6) (csc x) x x, x > x x, x > Â(ƒb : () sin hx ex e x (3) tan hx ex e x e x +

3 Â(ƒb5} : () (sin hx) cos hx () (cos hx) sin hx (3) (tan hx) sec hx (4) (cot hx) csc hx (5) (sec hx) sec hx tan hx (6) (csc hx) csc hx cot hx Â(ƒb5} : () (sin h x) + x, x R () (cos h x) x, x > (3) (tan h x) x, x < (4) (cot h x) x, x > (5) (sec h x) (6) (csc h x) x x, 0 < x < x + x, x > 0 3- D f(x) (g(x)) n (A) Df(x) n(dg(x)) n (B) g(x)df(x) nf(x)dg(x) (g(x)) n+ (C) Df(x) Dg(x) (g(x)) n (D) J,îÝ («) D f(x) (g(x)) Df(x) (g(x))n n(g(x)) n Dg(x) f(x) n (g(x)) n 3

x > (5) (sec h x) (6) (csc h x) x x, 0 < x < x + x, x > 0 3- D f(x) (g(x)) n (A) Df(x) n(dg(x)) n (B) g(x)df(x)

4 g(x) Df(x) nf(x) Dg(x) (g(x)) n+ (B) 3- J f(x) ÑªLƒb/ f (x) 3 f 3 (x) +, t (f ) (x) ( + x 3 ) 3 I y f(x), (f ) (y) f (x) 3 f 3 (x) y 3 (f ) (x) ( + x 3 ) J f(x) x 3 x, (f ) (6) I y f(x) x 3 x, y 6 x (f ) (6) f () 3 éæ J f(x) x 3 75, (f ) ( ) p: d(sin x) x (67 x) I y sin x, x (, ), y ( π, π ) sin y x, cos y dy dy cos y d sin y d x 4

3 x, (f ) (6) I y f(x) x 3 x, y 6 x (f ) (6) f () 3 éæ J f(x) x 3 75, (f ) ( ) 48 3-4

5 ÄÑ y ( π, π dy ), FJ cos y > 0,,ªŸA sin y x ¹ d sin x x éæ.f(x) sin x, f (x) Ê-µ_ Èì: (7 x) (A) π x π (B) π < x < π (C) x (D) < x <. p d(tan x) + x 3. p d(sec x) x x.(d) 3-5 Jì p: d (sin x) cos x (Ÿ ç5) d sin(x + h) sin x (sin x) lim h 0 h lim h 0 sin x cos h + cos x sin h sin x h lim[sin x cos h + cos x sin h h 0 h h ] cos h sin h sin x lim + cos x lim h 0 h h 0 h sin x 0 + cos x cos x éæ d(cos x) sin x 5

6 3-6 d (sinx ) 360 π, π 80 () (Ÿ) x π 80 x () d (sin x ) d sin( π 80 x) π 80 cos( π 80 x) π cos x q y ln(ln x), dy? () ln x () x (3) x ln x (4) J,îÝ (74ùx) y ln(ln x) dy d (ln x) ln x x ln x (3) 3-8 d ln sec x + tan x (A) sin x (B) cos x (C) cot x (D) csc x (E) sec x (7ãë) Ÿ d(sec x + tan x) sec x + tan x sec x + tan x (sec x tan x + sec x) sec x (E) 6

7 3-9 J y 6 x ( dy ) x (A) 8 (B) (C) 8 (D)0 (E) J,îÝ dy 6 x ( x ) ( dy ) x 4 ( ) 8 (A) 3-0 J f(x) x + x + 3x, f (3) f (x) (x + x + 3x) [ + (x + 3x) 3 ( + 3x )] x + x + [ + 3x x + 3x ( + 3 3x )], x 0 (70«çÍ) f (3) éæ. J f(x). J f(x) x + x + 3x + 9, f () (67«) x + x + x, f () (69«) () () J f(x) (x + + x ), + x f (x) (A) f(x) (B) f(x) (C) f(x) f (x) (x + + x ) ( + x + x ) (D) f(x) (E) [f(x)] 7

8 (x + + x ) x + + x + x f(x) + x + x f (x) f(x) (C) 3-. q m Ñcb, / f(x) (x + + x ) m, f (0) (A) (B) 0 (C) m (D) m (E)m.,æ f (0) (A) (B) 0 (C) m (D) m (E)m. f (x) m(x + + x ) m ( + x + x ) m + x (x + + x ) m f (0) m. f (x) m + x f(x) f (x) m (x + + x ) m m + ( + x ) f (x) 3 + x f (0) m ()(C), ()(E) éæ q a Q +, f(x) (x + x + ) a, p ( + x )f (x) + xf (x) a f(x) 0 (À ç5) 3-3 ûƒb f (x) () f(x) x + x + x () f(x) x(x + ) 8 (7 Ÿ 5)

9 ()f (x) x + x + [ + x x + x ( + x )] ()f (x) (x 3 + x ) 3 x + x (3 x + x ) éæ f(x) x + x 4 +, f (x) f (x) ( x + x 4 + ) ( x + (x4 + ) x 3 ) 3-4 f(x) + x, g(x) + f(x), f (x), g (x) (64A ç5) f(x) x + x, g(x) + x +x f (x) ( + x) x + x g (x) ( + x) 3-5 d (x + ( )4 + ) («55û) x + Ÿ [(x + )4 + ] 4(x + ) 3 (x + ) x (x + ) 4 + (x + ) (x4 + x 3 x ) (x + ) (x + ) y x x 4 x + 4 x 4, y (> û F) 9

10 y ( x x 4) x + 4 (x 4) x x 4 + x + 4 x (x + x 4 6) y 4 (x + (x4 6) 4x 3 ) (x + x 3 x4 6 ) Å ælø} Ü $(y}œà 3-7 J f(x) x + 3x + x, f () («67 ) â f (x) 3x + x (x + ) (3x + x) (6x + ) 3x + x x (3x + x) 3, x 0, 3 ) f () f(x) (x3 + ) + x + + x, f () («69 ) â ( + + x )f(x) (x 3 + ) + x siú x }, ) x + x f(x) + ( + + x )f (x) 3x + x + x(x3 + ) + x () 0

(6x + ) 3x + x x (3x + x) 3, x 0, 3 ) f () 3 4 3 3 6 3-8 f(x) (x3 + ) + x + + x, f () («69 ) â

11 f() + ( ) J x Hp () ( ) + ( + ) f () 3 + cü) f () Ç si úb) ln f(x) ln x ln + x ln + + x f (x) f(x) 3x x x + x + + x x + x f (x) (x3 + ) + x + ( 3x + x x x + x x ( + + x )( + x ) ) f () 3-9 q y (3x + ) 3 (x ) 4 (x 4) A 5 y y ( 3x + + B x + C x 4 ) v, A + B + C (A)0 (B) (C) (D)4 (E) 4 si úb, ) ln y 3 ln 3x + 4 ln x 5 ln x 4 y y 9 3x + + y 9 y( 3x x + 8 x + 5 x 4 5 x 4 ) A + B + C

12 3-0 ûƒb f (x) : ()f(x) (x3 + ) + x + x ()f(x) (x + ) 3 (x 3 + x + ) 4 (x 5 + x 3 + ) 6 (7Ÿ ) () si úb, ) ln f(x) ln(x 3 + ) + ln( + x ) ln( + x) f (x) f(x) 3x x x + x + x x ¹ f (x) (x3 + ) + x + ( 6x x x x + x x + x ), x > 0 ()ln f(x) 3 ln(x + ) + 4 ln(x 3 + x + ) + 6 ln(x 5 + x 3 + ) f (x) f(x) 3 x x x + x 3 + x x 4 + 3x x 5 + x 3 + ¹ f (x) (x + ) 3 (x 3 + x + ) 4 (x 5 + x 3 + ) 6 6x ( x + + x + 4 x 3 + x x4 + 8x x 5 + x 3 + ) éæ. y (x ) (x + ) 3 (x 3) 4, dy. y 5 (x ) x +, dy. (x + ) (x + ) 3 ( (x 3) 4 x + 3 x + 4 x 3 ). 5 (x ) x + [ 5 ( x x x + )] 3- q f(x) x( + x)( + x) (n + x) ( x)( x) (n x), f (x) (, RB) si úb) ln f(x) ln x + ln + x + ln + x + + ln n + x ln x ln x ln n x

3 + ¹ f (x) (x + ) 3 (x 3 + x + ) 4 (x 5 + x 3 + ) 6 6x ( x + + x + 4 x 3 + x + + 30x4 + 8x x 5 + x 3 + ) éæ. y (x ) (x + ) 3 (x 3) 4, dy. y 5 (x ) x +, dy.

13 f (x) f(x) x + + x + + x + + n + x + x + x + + n x f (x) x( + x)( + x) (n + x) ( x)( x) (n x) [ x + + x + + x + + n + x + x + x + + n x ] 3- q y Π n i (x a i) b i, y (Ÿ) si úb) ln y Σ n ib i ln x a i y y Σn i b i x a i y (Π n i (x a i) b i )(Σ n i b i (x a i ) ) ƒb f(x) x 3x + íûƒb («) (x 3) 3 () æbg<ƒbê/<õ.ª} I f(x) x 3 3x + (x 3) 3 si úb ln f(x) ln x + ln 3x + 3 ln x 3 3 siú x }) f (x) f(x) x + 3x + 6 x 3 3

b i x a i y (Π n i (x a i) b i )(Σ n i b i (x a i ) ) 3-3 3 ƒb f(x) x 3x + íûƒb («) (x 3) 3 ()

14 f (x) x 3 3x + (x 3) 3 ( x + 3x + 6 x 3 ), x 3, x 3 () ç x 3 v, f (x).æê (3) ç x 3 v, f (x).æê 3-4 q y ( + x)( + x ) ( + x 3 ) 3, O x3 ln y ln( + x) + ln( + x ) + 3 ln( + x3 ) Ä y y + x + x + x + 3 3x + x 3 y ( + x)( + x ) ( + x 3 ) 3 [ + x + x + x + x + x 3 ] éæ f(x) x x + 5 x + 7 (x + 3)(x + 4), f (x) x x + 5 x + 7 (x + 3)(x + 4) [ x + (x + ) + x 5(x + 7) x + 3 x + 4 ] 3-5 J y ln + x + x + x x, y y ln + x + x + x x ln + + x x ln( + x ) ln x y + x x x x x + x ( + x ) x( + x ) x x x 4

15 3-6 D x ln x + x (A) x x + (B) x (x ) (C) x x 4 (D) x x 4 (7û») Ÿ D x ln x + x D x[ln(x + ) ln(x )] [ x x + x x ] x x + x x x x 4 (D) 3-7 f(x) x tan x, f (x) (\ $ûf) f (x) tan x + x x + x 4 tan x + x + x y x sin x, y (>, \ û F) y x sin x x sin x cos x (sin x) x sin x x cos x sin 3 x éæ q y cos, dy x (A) 3 x sin x (B) sin (C) 3 x x sin x (D) sec x (E) sec x (65ãë) (C) 5

$x 4 tan x + x + x 4 3-8 y x sin x, y (>, \ û F) y x sin x x sin x cos x (sin x) x sin x x cos$

16 3-9 ø cos x + cos 3x + + cos(n )x sin nx sin x, sin x + 3 sin 3x + + (n ) sin(n )x? cos x + cos 3x + + cos(n )x sin nx sin x si vú x }), [sin x + 3 sin 3x + + (n ) sin(n )x] sin x n cos(nx) cos x sin(nx) ( sin x) 4n sin x cos(nx) cos x sin(nx) 4 sin x sin x + 3 sin 3x + + (n ) sin(n )x [4n sin x cos(nx) cos x sin(nx)] 4 sin x 3-30 f(x) x sin x + x, f (x) (µ± ç5) f (x) sin x + x x x x sin x éæ y cos( π e x ), dy x0 (A) π (A) (B) π (C) 0 (D) (E) e (7ãë) 3-3 q f(x) sin x b a b, b < x < a, f (x) (µ± ç5) f (x) x b a b (a x)(x b) a b x b 6

sin x cos(nx) cos x sin(nx) 4 sin x sin x + 3 sin 3x + + (n ) sin(n )x [4n sin x cos(nx) cos x sin(nx)] 4 sin x 3-30 f(x) x sin x + x,

17 x 3-3 Compute clerivative :D sin(x + ) (65«ı) éæ y y x D sin(x + ) sin(x + ) x cos(x + ) sin (x + ) sin x + cos x, y (70«ç5) + cos x 3-33 l- :() d sec x () d sec x () desin x sec( x ) tan( x ) x (69ÀM ç5) () desin x e sin x x 3-34 y tan (sin hx) (\ û F) y cos hx + (sin hx) cos hx 3-35 J y log x (sin x), x (0, ), dy æûl²: y log x (sin x) ln(sin x) ln x (A ) dy cos x ln x ln(sin x) sin x x (ln x) cot x ln x ln sin x x (ln x) x cot x ln x ln sin x x(ln x) 7

$3-34 y tan (sin hx) (\ û F) y cos hx + (sin hx) cos hx 3-35 J y log x (sin x), x (0, ), dy æûl²: y log x (sin x)$

18 3 éæ x y log 7 + 5, x y y log 7 3 5x + 5 x log q y ln(e x + e x + ), y y e x + e x + e x (ex + e x + ) e x + e x + ex ( e x + + e) e x + e x e x + éæ y ln(x + a + x + ax + b ), y x + ax + b 3-37 ƒb y e ln x+x, y k: (A) e x (B) e x ( + x) (C) e x ( x) (D) xe x Å (7û») à e ln x x Ä y e ln x+x e ln x e x x e x ) y e x + xe x e x ( + x) (B) 3-38 J x > 0, d dt (xt ) (A) t x t (B) x t ln t (C) x t ln x (D).æÊ (> $û) Å d dt (xt ) d dt (et ln x ) e t ln x ln x x t ln x (C),æJ d xt k tx t 8

(C) e x ( x) (D) xe x Å (7û») à e ln x x Ä y e ln x+x e ln x e x x e x ) y e x + xe x e x ( + x) (B) 3-38 J x > 0, d dt (xt

19 3-39 J f(x) x x, x > 0, f (x) Ñ (A)x x (B)x x ln x (C)x x (x + ln x) (D)x x ( + ln x) (7 x, «) Ä f(x) x x e x ln x ) f (x) e x ln x (ln x + x x ) xx ( + ln x) (D) éæ y x 0 0 x, y (56«ç5) y 0x 9 0 x + x 0 0 x ln J y (ln x) ln x, dy (70RB ç5) Ä y ln x ln x ln x ln ln x e ) dy eln x ln ln x ( x ln ln x + ln x x ln x ) (ln x) ln x [ln(ln x) + ] x (ln x) ln x [ln(ln x) + ] x éæ Find f (e),where f(x) (ln x) ln x (7µ± ç5) e 3-4 J y x x, x > 0, dy (A ) Ä y x x e x ln x 9

(70RB ç5) Ä y ln x ln x ln x ln ln x e ) dy eln x ln ln x ( x ln ln x + ln x x ln x ) (ln x) ln x [ln(ln x) + ] x (ln

20 ) dy x e x ln ( x x x + x ln x) xx ( x + x ln x) xx x ( + ln x) x x ( + ln x) éæ y x x, dy x x ( x ln x x ) 3-4 q y sin(cos x) cos(sin x), dy dy cos(cos x ) cos x ( sin x) cos(sin x)+sin(cos x) [ sin(sin x)] sin x cos x sin x cos x[cos(cos x) cos(sin x) + sin(cos x) sin(sin x)] sin x cos(cos x sin x) sin x cos(cos x) 3-43 q y sin (x x ), y y 4x ( x ) ( x x + x x ) ( x ) ( x ) x x ( x ) x x 0

sin(sin x)] sin x cos x sin x cos x[cos(cos x) cos(sin x) + sin(cos x) sin(sin x)] sin x cos(cos x

21 Ä ç x < v, y x ç < x < v, y x 3-44 J y sin (tan hx), dy dy (tan hx) sec hx (Ÿ66 ç5) (sec hx) sec hx sec hx 3-45 sin h x Ñ (A) ln(x + x + ) (B) ln(x x ) (C) ln x + x (D) ln( x + x) («) I y sin h x, sin hy x Yì ey e y x (e y ) xe y (e y ) xe y 0 e y x ± 4x + 4 x ± x + (Š. ) Ä e y x + x + ¹ y ln(x + x + ) (A) 3-46 D x sin h x Ñ (A) x + (B) x + (C) x (D) x («)

22 I y sin h x, sin hy x siú x }, ) cos hy dy dy cos hy + sin hy + x (A) 3-47 If y (ln x) x, x >,then dy is equal to (A) (ln x)x [ + ln ln x] ln x (B) (ln x) x [ ln x + ln x] (C) (ln x)x [ + ln x] ln ln x (D) x(ln x)x (E) none of them (70Ÿ ç5) Ä y (ln x) x x ln ln x e ) dy ex ln ln x (ln ln x + x x ln x ) (ln x) x (ln ln x + ln x ) (A) éæ y (x + ) ln x, y (70RB ç5) y (x + ) ln x [ ln(x + ) x + x ln x x + ] 3-48 y (log x) log x, y (log x log 0 x) (RB)

23 < æj0ñ~d3-3ªœ Ä d log x d ln x ln 0 ln 0 x y (log x) log x log x ln log x e ) y e log x ln log x ( x ln 0 ln log x + log x log x x ln 0 ) (log x) log x (ln log x + ) x ln () } 3 sin x + cos 3x («) () } 3 3x + x («, A ) () Ä 3 sin x + cos 3x e sin x ln 3 cos 3x ln + e ) d (3sin x + cos 3x ) e sin x ln 3 ( cos x ln 3) + e cos 3x ln ( 3 sin 3x ln ) 3 sin x ( cos x ln 3) cos 3x (3 sin 3x ln ) () Ä 3 3x + x e 3x ln 3+ + e x ln ) d + (33x x ) e 3x ln 3 [3 x (ln 3) ] + e x ln (x ln ) 3 3x [3 x (ln 3) ] + x (x ln ) 3-50 q y x aa + a xa + a ax, a > 0, x > 0, dy Ä d ) a a x (xaa aa (66Ÿ ç5)( ) 3

24 d ) d ln a (axa exa e xa ln a (ax a ln a) a xa (ax a ln a) d ) d ln a (aax eax e ax ln a (a x (ln a) ) a ax (a x (ln a) ) ) dy aa x aa + a xa (ax a ln a) + a ax [a x (ln a) ] éæ d (aax ) (A) a ax (B) a x a ax (C) a x a ax (ln a) (D) a x a ax ln a (E) J,îÝ («) (C) 3-5 J y x xa + x ax + a xx, a > 0, x > 0, dy («, A ) Ä d ) d ln x ) (xxa (exa e xa ln x (ax a ln x + x a x ) x xa x a ( + a ln x) d ) d ln x ) (xax (eax e ax ln x (a x ln a ln x + a x x ) x ax a x ( + ln a ln x) x d ) d ln a ) (axx (exx 4

25 e xx ln a ( d (xx ) ln a) a xx x x [ + ln x] ln a ) dy xxa x a ( + a ln x) + x ax a x ( x + ln a ln x) + axx x x [ + ln x] ln a 3-5 J y ( + a x )bx, a > 0, b > 0, dy Ä y e bx ln(+ a x ) (A ) ) dy a ln(+ ebx x ) [b ln( + a x ) + bx + a x ( + a x )bx [b ln( + a x ) ab x + a ] a x ] 3-53 q y x x, y () (66ùx, A ç5) Ä y x x e x ln x ) dy ex ln x (x ln x + x x ) x x (x ln x + x) ] y () 3-54 q y (ax + bx + c) x, y Ä y e x ln(ax +bx+c) ) y e x ln(ax +bx+c) [ln(ax + bx + c) + (ax + bx + c) x [ln(ax + bx + c) + 5 x(ax + b) ax + bx + c ] x(ax + b) ax + bx + c ]

26 3-55 J x > 0, t D x x sin x («) Ä x sin x sin x ln x e ) D x x sin x sin x ln x D x e e sin x ln x (cos x ln x + sin x x ) x sin x (cos x ln x + sin x x ) éæ () y (sin x) sin x, y () y (sin x) tan x, y () y (sin x) sin x cos( + ln(sin x)) () y (sin x) tan x [ + sec x ln(sin x)] 3-56 J f(x) x (xx), f () (70«) x > 0, x (xx) e xx ln x d (x(xx) ) e xx ln x [ d (xx ) ln x + x x x ] x (xx) [x x ( + ln x) ln x + x x ] f () 64 [(ln ) + ln + ] 3-57 y x e x, dy (A ) dy x ln x e ex x x (e x ) x (ln x) e x 6

27 3-58 q g(x) x x + log (log x), dg ( ½) Ä log (log x) ln( ln x ln ) ln [ln(ln x) ln ln ] ln ] dg xx ( + ln x) + x ln x ln 3-59 q f(x) x ln x + a ln x + a x, f (x) f (x) ln x + x x + a x + ax ln a + a x + ln x + ax ln a 3-60 y e ex + x sin x + log 3 (log 5 x), y log 3 (log 5 x) ln(log 5 x) ln 3 ln 3 ln(ln x ln 5 ) (ln ln x ln ln 5) ln 3 y e ex e x + ( )( x ) ( x) sin x + x e ex +x + x sin x x + x ln x ln 3 x + x ln x ln 3 7

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