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1

2

3 x n e x sin x, cos x

4

5 5 100%!! 1

6 6...

7 y = f(x) dx dx := lim h 0 f(x + h) f(x) h

8 8 1 f (x) y = f(x) f (x) y = f(x) *1 1.2 (linearity) {af(x) + bg(x)} = af (x) + bg (x). 1.3 x n (x n ) = nx n 1. x 3, x 5 ( ) 1 ( ) *1

9 1.3 x n 9 x 1, x 1 3 OK (C) = y = 2x 2 3x + 1 (x n ) = nx n 1, (C) = 0 y = (2x 2 3x + 1) = 2(x 2 ) 3(x) + (1) = 2 2x = 4x y = x x + x x. x (x n ) = nx n 1 y = x x + x x. = x x x 3 2 y = 1 2 x x x 1 2 = 1 2 x 1 3x 3 x x.

10 e x Napier e = (e x ) = e x. 1.5 sin x, cos x cos sin cos sin (sin x) = cos x, (cos x) = sin x. 1.6 e log e x log e x log x (log x) = 1 x....

11 (x n ) = nx n 1 (sin x) = cos x (cos x) = sin x (e x ) = e x (log x) = 1 x...., f(x) g(x) *2 { f(x)g(x) } = f (x)g(x) + f(x)g (x). *2 { f(x)g(x) } = f (x)g (x)!!

12 12 1 f (x) g(x) + f(x) g (x) 3. y = x log x.!! y = x f(x) y = (x) log x g(x) log x + x = 1 log x + x 1 x (log x) = log x y = e x sin x. e x sin x y = e x f(x) sin x g(x) y = (e x ) sin x + e x (sin x) = e x sin x + e x cos x = e x (sin x + cos x).

13 y = x sin x cos x. x, sin x, cos x f(x)g(x) h(x) { f(x)g(x)h(x) } = { f(x)g(x) } h(x) + f(x)g(x) h (x) = { f (x) g(x) + f(x) g (x) } h(x) + f(x)g(x)h (x) = f (x) g(x)h(x) + f(x) g (x) h(x) + f(x)g(x) h (x). 3 { f(x)g(x)h(x) } = f (x) g(x)h(x) + f(x) g (x) h(x) + f(x)g(x) h (x). y = (x) sin x cos x + x (sin x) cos x + x sin x (cos x) = sin x cos x + x cos 2 x x sin 2 x.

14 !!!! { f(x) g(x) } = f (x)g(x) f(x)g (x) { g(x) } y = 2x + 1 x

15 y = (2x + 1) (x 2 + 3) (2x + 1)(x 2 + 3) (x 2 + 3) 2 = 2(x2 + 3) (2x + 1) 2x (x 2 + 3) 2 = 2x x 2 2x (x 2 + 3) 2 = 2x2 2x + 6 (x 2 + 3) (tan x) = 1 cos 2 x. (sin x) = cos x, (cos x) = sin x 2 tan tan x tan tan tan x = sin x cos x (tan x) = ( ) sin x. cos x (tan x) = cos 2 x + sin 2 x = 1 ( ) sin x = (sin x) cos x sin x(cos x) cos x cos 2 x = cos2 x + sin 2 x cos 2. x (tan x) = cos2 x + sin 2 x cos 2 x = 1 cos 2 x.

16 tan tan x ( )

17 y = (2x + 1) 7. (i) 7 (ii) y = 128 x x x x x x x + 1 y = 896 x x x x x x + 14 (1) 2x + 1 u = 2x + 1 y = 2x + 1 u y = u 7. (2) y u!! u (3) du du = 7u6. 7 dx (u) du dx = (2x + 1) = 2 y = dx y = dx = du du dx = 7u6 2 = 14u 6 = 14(2x + 1) 6.

18 18 1 u u dx = du du dx u u. 8. y = e 2x+3. dx = du du dx u = 2x + 3 y = e u u du = eu. du dx = 2. dx = du du dx = eu 2 = 2e 2x y = sin (4x + 1) u = 4x + 1 y = sin u du = cos u.

19 dx = du du dx du dx = 4. = cos u 4 = 4 cos (4x + 1). 10. y = log (3x 5) u = 3x 5 y = log u du = 1 u. du dx = 3. dx = du du dx = 3 3x 5. ( e 2x+3 ) = 2e 2x+3 { sin (4x + 1) } = 4 cos (4x + 1) { log (3x 5) } = 3 3x 5. u = px + q( ) ( e px+q ) = p e px + q { sin (px + q) } = p cos (px + q) } {{ } { log (px + q) } 1 = p px + q

20 20 1 x { f(px + q) } = pf (px + q). u { cos (3x + 7) } = 3 { sin(3x + 7) } = 3 sin (3x + 7). { tan (5x) } 1 = 5 cos 2 5x = 5 cos 2 5x. ( e 3x+1 ) = 3e 3x+1. f(px + q) = pf(px + q)!!!! u u u... f(px + q) u 11. y = 1 x u. u = x y = 1 u = u 1. u dx = du du dx = 2x u 2 2x = (x 2 + 1) 2. x ( ) (u ). y = 1 x ( = x ) 1

21 ( y = ) 2 x } {{ } ( d = du) x ( d dx = du dx ) 1 = (x 2 + 1) 2 2x = 2x (x 2 + 1) 2. u 12. y = sin { cos (tan x) } u { } y = sin cos (tan x) { y = cos cos (tan x) } cos (tan x) } {{ } ( ) cos (tan x) = sin (tan x) } {{ } cos (tan x) } {{ } (tan x) sin (tan x) = cos 2 x y cos { cos (tan x) } sin (tan x) = cos 2 x y = sin { cos (tan x) }

22 22 1 (1) sin (2) cos (3) tan x (4) 1.10 (x n ) = nx n 1. (e x ) = e x. (sin x) = cos x. (cos x) = sin x. (log x) = 1 x. { f(x)g(x) } = f (x)g(x) + f(x)g (x). { f(x) g(x) } = f (x)g(x) f(x)g (x) { g(x) } 2. dx = du du dx. ( )

23 Good Luck!!

24

25 (e x ) = e x, (log x) = 1 x. 2.1 * 1 *1 y = f(x) x y y x x y x = g(y) y = g(x) f(x) g(x) f 1 (x) y = f(x) x x y x f(x) y y x f(x) y y = f(x) x y g(y) x y x x = g(y) y = g(x) f 1 (x) f y = x 2

26 26 2 x log x e } log {{ x = x }!! x x = e log x e x = ±2 y = x 2 y y = 2 2 = 4, y = ( 2) 2 = 4. x y = 4 x = 2 y = x 2 y = 4 x = 2 y x y = 4 y = 4 x = 2 x = 2, y = x 2 x ( ) y = x 2 (x 0) x 4 2!! 1 y = f(x) 1!! x y x 1 x 2 f(x 1 ) f(x 2 ) ( x y ) f(x 1 ) = f(x 2 ) x 1 = x 2 ( ) y = f(x) (injection) f 1 (f(x)) = x, f ( f 1 (x) ) = x f 1 f(x) (1) x (2) = x (3)

27 y = 2 x. y = e x x = e log x y = 2 x = e log 2x ( 2 x e!!) = e x log 2 (log log a b = b log a). e x log 2 x ( e 2x) = 2e 2x y = log 2 e x log 2 log 2x = log 2 } e {{} = 2 x log 2. 2 x 14. (a x ) = a x log a. a x log ax = e = e x log a (a x ) = ( e x log a) = log a e x log a = log a e log ax a x = a x log a.

28 ( 1)... ( 2)!? (Step1) y = 2 x (Step2) (Step3). (Step4) a (Step5) 2 (Step1) a (a x ) = a x log a e

29 !! (e...!!) y = log 2 x. 2 ( e ) (log x) = 1 x. *2 log a b = log c b log c a. e y = log 2 x e y = log 2 x = log x log 2. *2 x = log a b a a x = a log a b = b. log a x a x c log c a x = log c b x log c a = log c b x = log c b log c a.

30 30 2 y 1 log 2 y = ( ) log x = 1 log 2 log 2 (log x) = 1 x log 2. (log a x) = 1 x log a.... (log a x) = ( ) log x = 1 log a log a (log x) = 1 x log a. 2.3 e.g. y = x x *3, x... *3 x x 1 x x x 1... x

31 log log a b = b log a log y = log x x log y = x log x. x y x *4 1 y dx y y = log x + 1 y = x x dx dx dx y dx = y(log x + 1). y = x x dx = xx (log x + 1). *4 y = (2x + 1) 10 u = 2x + 1 u 10 x 10u 9 2 u u log y x

32 32 2 (i) y = f(x)!? (ii) log y = log f(x) (iii) x 1 y dx = (iv) ( y ) dx x dx 1 e.g. y = x cos x * 5 *5 x... log

33 log log y = log x cos x log y = cos x log x x 1 y dx = sin x log x + cos x 1 x. y ( dx = y sin x log x + cos x 1 ). x 2.4

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

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