1109291212434690_1212270774412.pdf

: Derivative of Function 1

-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::

[ ]

L P

LO

P

P (Tangent Line) P

P L P

L

0

Y

X

P

y = f(x) P(a, b) Q(a + h, b + k)

h 0

0X

Y

b+k

a+h

b

a

P(a, b)

Q(a+h,b+k)

PQ PQ (Secant Line)

( )( )b k b ka h a h

+ =+ b + k = f(a + h) b = f(a)



= ( ) ( )f a h f a

h

+

kh =

( ) ( )f a h f ah

+

Q1 P Q PQ1 Q2 P Q1 PQ2 Q3 P Q2 PQ3 Qn PQn Qn P Qn P

PQn P

P ( ) ( )

h 0

f a h f alim

h+

y = f(x) P(x, y) P

( ) ( )

h 0

f a h f alim

h+

()

P(x, y) P

1 3

yx

= (3, 1) f(x) =

3x (3, 1)

( ) ( )

h 0

f 3 h f 3lim

h+

= h 0

31

3 hlimh

+ =

h 0

hlim

h(3 h)+

= h 0

1lim

(3 h)+ =

13

(3, 1) 13

(Tangent Line and Normal Line)

PQ P(x1, y1) m PQ y y1 = m(x x1)

PS PQ

PS y y1 = 1m

(x x1)

Q

S

P(x1,y1)

0

Y

X

X

Y

0

P

Q4Q3

Q2Q1



2 y = 2x x2 (1, 1) 2.1 2.2 2.3

f(x) = 2x x2 (1, 1)

( ) ( )

h 0

f 1 h f 1lim

h+

= ( ) ( )2

h 0

2 1 h 1 h 1lim

h

+ +

= ( )2

h 0

2 2h 1 2h h 1lim

h+ + +

= 2

h 0

hlim

h

= h 0lim h

= 0 (3, 1) 0 m = 0 y 1 = 0(x 1) y 1 = 0 y 1 = 0 m = 0

y 1 = 10(x 1)

x 1 = 0 x 1 = 0

1.

2. y 2 x 1 = 0 y

1 1) P

1. y = 2x2 3x P(1, -1) 2. xy = 2 P(-1, -2)

3. y = 2x 1x+

P(1, 2)

4. y = 5 + 4x 3x2 P(3, -10) 5. y = x x = 9

2) y = ax y = 3x2 + 8 (1, 11) a



y = f(x) P(a, b) Q(a + h, b+ k)

( ) ( )f a h f a

h

+

y

x x a a + h yx

Q P h

( ) ( )

h 0

f a h f alim

h+

x = a

y = f(x) a f y x x

a a + h ( ) ( )f a h f a

h

+

y x x = a ( ) ( )

h 0

f a h f alim

h+

1 y x x = 3 x = 3.2

y = f(x) = x2 2x + 5

yx

=

( ) ( )f x h f xh

+

= ( ) ( )f 3 0.2 f 3

0.2

+ =

8.84 80.2

= 0.840.2

= 4.2

4.2 2 y = 2x2 3 y x x = 2 x = 2

= ( ) ( )

h 0

f x h f xlim

h+

= ( )2 2

h 0

2 x h 3 2x 3lim

h+ +

= h 0lim 4x 2h + = 4x

x = 2 y x 4(2( = 8 y x 8

0X

Y

b+k

a+h

b

a

P(a, b)

Q(a+h,b+k)



3 3.1 5 8 3.2 3.3 5

x y y = (x)(x) = x2 3.1 a + h = 8 a = 5 h = 3

yx

=

( ) ( )f x h f xh

+

= ( ) ( )f 5 3 f 5

3

+

= 64 25

3

= 13 /

3.2

( ) ( )h 0

f x h f xlim

h+

= ( )2 2

h 0

x h xlim

h+

= h 0lim 2x h +

= 2x 3.3 5

5.2 2(5) = 10 /

2 1) y x

1. y = x2 3x x = 2 x = 1.8 2. y = x3 + x2 x = - 2.5 x = 2.2 3. y = 2x2 + 4x + 1 x = 0 x = 0.001 4. y = 3x2 6x + 5 x = 1.2 x = 1.1 5. y = x3 2x2 x + 1 x = 1 x = - 3

2) y = f(x) f(1) = 2 y x 1 3 7 x = 3 y

3) y = f(x) f(1) = 0, f(8) = 8 y x 1 5 6 x = 5 y

4) 2 5) v r

v = 34

r3

5.1 6 9

5.2 9



6) t Q

t Q t

Q 1210

= 6.1 t = 0 t = 10 6.2 t = 10

7) 10 12 7.1 7.2 10

8) 10 9 8.1 8.2 10

9) N t 8Nt 1

= + t N t t = 3

10) 400 15 PV = 6000 ( P V ) P V V = 100

11) 11.1 11.2

12) s r 2k

rs

= k > 0 r s s

y = f(x) f(x) f(x) = ( ) ( )f x h f x

h

+ h 0

f(x) f x y = f(x) f x

f(x) = ( ) ( )h 0

f x h f xlim

h+

y = f(x) x = x0 ( ) ( )

h 0

f x h f xlim

h+

f

(differentiable) x0 f f (a, b)

( ) ( )

h 0

f x h f xlim

h+

f x0 f x0

f x



y = f(x) x f(x) y

dydx

( d y d x)

dfdx

d

f(x)dx

f(x)

y = f(x) x = a

x a

y = x a

dydx =

x a

df(x)

dx =

x a= (Evaluating Symbol) x = a

2

1. 2. x = a

1 f(x) = 2x2 x

f(x) = ( ) ( )h 0

f x h f xlim

h+

= ( ) ( )2 2

h 0

2 x h (x h) 2x xlim

h

+ +

= 2 2 2

h 0

2x 4xh 2h x h 2x xlim

h+ + +

= 2

h 0

4xh 2h hlim

h+

= h 0lim 4x 2h 1 +

= 4x - 1 4x 1 2 f(x) = x

f(x) = ( ) ( )h 0

f x h f xlim

h+

= h 0

x h xlim

h+

= ( ) ( )

( )h 0x h x x h x

limh x h x

+ + ++ +

= ( )h 0 x h xlim h x h x + + +

Q

S

P(x1,y1)

0

Y

X



= ( )h 0 hlim h x h x + + =

h 0

1lim

x h x + + = 1

2 h

1

2 h

3 y = f(x) = 1x

. f x 0 . y = 1

x x = 2

. x 0 f(x) = ( ) ( )

h 0

f x h f xlim

h+

= h 0

1 1x h xlim

h

+

= h 0

1 1 1lim

x h x h +

= ( )

h 0

x x hlim

h(x h)x +

+

= h 0

1lim

(x h)x+

= 1

(x 0)x+ = 2

1x

21x

. y = 1x x = 2

f(2) = 14

y = f(2) = 12

y = 1x (2,

12

)

y - 12

= 14

(x 2) x + 4y 4 = 0 x + 4y 4 = 0



4 f(x) = (x 1)2 x = 1

f(x) = ( ) ( )h 0

f x h f xlim

h+

= ( ) ( )2 2

h 0

x h 1 x 1lim

h+ +

= ( ) ( )

h 0

x h 1 x 1 x h 1 x 1lim

h+ + + + +

= ( )

h 0

h 2x h 2lim

h+

= ( )h 0lim 2x h 2 +

= 2x 2 f(1) = 2(1) 2 = 0 x = 1 f(1) = 0

3 1) y = f(x)

1. f(x) = 1x 3+

2. f(x) = 2 - x 3. f(x) = 3x2 + 1

4. f(x) = x 1x 1+

2) 2x ; x 2

f(x)2x ; x 2

= >

2.1 f(- 1) f(3) 2.2 f x = 2

3) 2x 1 ; x 1

f(x)5 ; x 1

+ = = g x = 1

4) 1yx 1

= (0, -1) 5) y = x2 1 (1, 0) (-2, 3)



1 f(x) = c c f(x) = 0 f(x) = ( ) ( )

h 0

f x h f xlim

h+

= h 0

c clim

h

= h 0lim 0

= 0 2 f(x) = x f(x) = 1 f(x) = ( ) ( )

h 0

f x h f xlim

h+

= h 0

x h xlim

h+

= h 0

hlim

h

= h 0lim 1

= 1 3 f(x) = xn n f(x) = nxn-1 f(x) = ( ) ( )

h 0

f x h f xlim

h+

= ( ) ( )n n

h 0

x h xlim

h+

=

n n 1 n 2 2 n n

h 0

n n n nx x h x h ... h x

0 1 2 nlim

h

+ + + +

=

( )n 1 n 2 2 nh 0

n n 1nx h x h ... h

2limh

+ + +

= ( )n 1 n 2 n 1

h 0

n n 1lim nx x h ... h

2

+ + +

= nxn-1



4 f g x (f + g)(x) = f(x) + g(x) F(x) = f(x) + g(x)

F(x) = ( ) ( )h 0

F x h F xlim

h+

= ( ) ( ) ( ) ( )

h 0

f x h g x h f x g xlim

h + + + +

= ( ) ( ) ( ) ( )

h 0

f x h f x g x h g xlim

h + + +

= ( ) ( ) ( ) ( )

h 0 h 0

f x h f x g x h g xlim lim

h h + + +

= f(x) + g(x) 5 f g x (f g)(x) = f(x) - g(x) F(x) = f(x) - g(x)

F(x) = ( ) ( )h 0

F x h F xlim

h+

= ( ) ( ) ( ) ( )

h 0


h + +

= ( ) ( ) ( ) ( )

h 0

f x h f x g x h g xlim

h + +

= ( ) ( ) ( ) ( )

h 0 h 0

f x h f x g x h g xlim lim

h h + +

= f(x) - g(x) 6 c f x (cf)(x) = c(f(x)) F(x) = cf(x)

F(x) = ( ) ( )h 0

F x h F xlim

h+

= ( ) ( )

h 0

cf x h cf xlim

h+

= ( ) ( )

h 0

f x h f xlim c

h +

= ( ) ( )

h 0

f x h f xc lim

h+

= c(f(x))



7 f g x (fg)(x) = f(x)g (x) + f(x)g(x) F(x) = f(x)g(x) F(x) = ( ) ( ) ( ) ( )

h 0


h+ +

= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

h 0

f x h g x h f x h g x f x h g x f x g xlim

h+ + + + +

= ( ) ( ) ( ) ( ) ( ) ( )h 0

g x h g x f x h f xlim f x h g x

h h + + + +

= ( ) ( ) ( ) ( ) ( ) ( )h 0 h 0 h 0 h 0

g x h g x f x h f xlim f x h lim lim g x lim

h h + + + +

= f(x)g (x) + f(x)g(x) 8 f g x

fg

(x) = ( ) ( ) ( ) ( )

( )( )2g x f x f x g x

g x

y = ( )( )

f x

g x

dydx

=

( )( )

( )( )

h 0

f x h f xg x h g x

limh

+ +

= ( ) ( ) ( ) ( )

( ) ( )h 0f x h g x f x g x h

limh g x g x h

+ + +

= ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )h 0f x h g x f x g x f x g x h f x g x

limh g x g x h

+ + + +

= ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )h 0f x h f x g x h h x

g x f xh h

limg x g x h

+ + +

= ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )h 0 h 0 h 0 h 0h 0 h 0

f x h f x g x h h xlim g x lim lim f x lim

h hlim g x lim g x h

+ + +

= ( ) ( ) ( ) ( )

( )( )2g x f x f x g x

g x



8 1 f(x) = c c f(x) = 0 ( )d c

dx = 0

2 f(x) = x f(x) = 1 ( )d x

dx = 1

3 f(x) = xn n f(x) = nxn-1 ( )nd x

dx = nxn-1

3.1 u x f(x) = un n f(x) = nun-1f(u) ( )nd u

dx = nxn-1 ( )d u

dx

4 f g x (f + g)(x) = f(x) + g(x) ( )d u v

dx+ = ( ) ( )d du v

dx dx+

5 f g x f(x) = u, g(x) = v (f g)(x) = f(x) - g(x) ( )d u v

dx = ( ) ( )d du v

dx dx

6 c f x f(x) = u (cf)(x) = c(f(x)) ( )d cu

dx =

dc udx

7 f g x f(x) = u, g(x) = v (fg)(x) = f(x)g (x) + f(x)g(x) ( )d uv

dx = ( ) ( )d du v v u

dx dx+

8 f g x f(x) = u, g(x) = v

fg

(x) = ( ) ( ) ( ) ( )

( )( )2g x f x f x g x

g x

d udx v

= ( ) ( )

2

d dv u u vdx dx

v



8 ( 1 8)

u, v w x c

1. ( )d uvwdx

= ( ) ( ) ( )d d duv w uw v vw udx dx dx

+ +

2. d udx c

= ( )1 d

uc dx

c

3. dydx

= 1dxdy

4. d xdx

= 1

2 x

5. d udx

= ( )1 d udx2 u

6. y = [f(x)]n dydx

= ( ) ( )n 1 dn f x f xdx

1 y = - 5

dydx

= ( )d 5dx

= 0 2 y = x5

dydx

= 5x5-1 = 5x4

3 f(x) = 8x3 2x2 + 5x - 7 f(x) = 8x3 2x2 + 5x 7

f(x) = ( )3 2d 8x 2x 5x 7dx

+ = ( ) ( ) ( ) ( )3 2d d d d8x 2x 5x 7

dx dx dx dx +

= 8(3x2) - 2(2x) + 5 = 24x2 - 4x + 5 4 y = (3x + 2)2

dydx

= 2(3x + 2)2-1 ( )d 3x 2dx

+ = 6(3x + 2) = 18x + 12



5 y = 2(5 2x2)3

dydx

= ( )32d2 5 2xdx

= ( ) ( )3 12 2d2 3 5 2x 5 2xdx

= ( ) ( )222 3 5 2x 4x = ( )2224x 5 2x

6 y = 31x

31x

= x-3

dydx

= - 3x-3-1 = - 3x-4 = 43x

7 y = x

x = 12x

dydx

= 12d x

dx

=

11

21 x2

=

121 x

2

=

1

2 x

13 4 8 y = x 1 x 1 = ( ) 12x 1

dydx

= ( ) 12d x 1dx

= ( ) ( )1 121 dx 1 x 12 dx

= ( ) 121 x 12

= 12 x 1



9 y = 2x 1

2x 1 = ( ) 12 2x 1

dydx

= ( ) 12 2d x 1dx

= ( ) ( )1 12 221 dx 1 x 12 dx

= ( ) 12 22x x 12

= 2

x

x 1

10 y = (x 2)(x + 3)

dydx

= ( ) ( ) ( ) ( )d dx 2 x 3 x 3 x 2dx dx

+ + + = ( ) ( ) ( ) ( )x 2 1 x 3 1 + + = 2x 1+

(x 2)(x + 3) = x2 + x 6

( )2d x x 6dx

+ = 2x + 1

11 y = 2x 12x 1

+

dydx

= d 2x 1dx 2x 1

+

= ( ) ( ) ( ) ( )

( )2d d

2x 1 2x 1 2x 1 2x 1dx dx

2x 1

+ ++

= ( ) ( ) ( ) ( )

( )22x 1 2 2x 1 2

2x 1

+ +

= ( )24x 2 4x 2

2x 1

+ ++

= ( )24

2x 1+



12 f(x) = 2x3 4x2 x f(x) = 0 f(x) = 2x3 4x2 f(x) = 6x2 8x f(x) = 0 6x2 8x = 0 2x(3x 4) = 0

x = 0, 43

x f(x) = 0 0, 43

13 y = x + x3 4 y = x + x3 y = 1 + 3x2 1 + 3x2 = 4 3x2 + 1 4 = 0 3x2 3 = 0 x2 1 = 0 (x 1)(x + 1) = 0 x = 1, - 1 1 1 y = x + x3 x = 1 y = 2 x = - 1 y = - 2 (1, 2) (-1, -2) 4 14 y = x x = 4 y = x x =4 y = 4 = 2 (4, 2)

y = 12 x

( )

x 4

y = = 1

2 4 =

14

14 (4, 2)

y 2 = 14(x 4)

4y 8 = x 4 x 4y + 4 = 0 x 4y + 4 = 0



4

1)

1. y 3= 2. 3 xy x

3= +

3. 3y x 3x 7= + 4. 3 2 2 3

1 2y x 3x x

x x= + +

5. 1y xx

= + 6. 2y 2x x= 7. ( )22y x 2x= 8. ( ) ( )y x x 1 x 2= + + 9. ( ) ( )2 2y 4x x x 3= + 10. 4 3 2 3

1 2 2 1y x x

4 3 x x= +

11. 3 2x 3x 5x 2

yx

+ = 12. y 1 x= + 13.

22x xy

x

+=

14. ( )53x 2y x 1x

+ = +

15. ( )7 2 x 1y 2x xx 1 = +

16. 3

1y

2 x= +

17. 3x 8y2x 5

+= +



2)

1 31

f(x) 2xx

= x = 1

2 5 3 21 1 1

f(x) x x x 4x 55 3 2

= + + x = 1

3 ( ) ( )2 2f(x) 2x 3x 1 x x= + x = - 1 4

2x 1f(x)

2x 1= +

x = 2

3) f(4) = 4 f(4) = - 5 g(4) 1. ( )y xf x= 2. ( )f xy

x=

4) y = f(x) = 1 x2 P(0, 2)

5) y = 3 3xx

+ x 6) 3 y = x2 x (a, b) a, b 7) (x0, y0) y = x2 + 1 (x0, y0) y = 6x + 8

x0 + y0

8)

23 23

2

2x 2x 3x 1f(x)

x + = (1, 2)

(1, 2)

(Transcendental Function)

1. n n 1d du

(u ) nudx dx

=

2. d 1 dv

ln vdx v dx

= 3. aa log ed dvlog vdx v dx= ; a > 0 a 1 4. v v

d dve e

dx dx= 5. v vd dva a ln a

dx dx= ; a > 0 a 1

1. d dv

(sin v) cos vdx dx

= 2. d dv(cos v) sin vdx dx

= 3. 2

d dv(tan v) sec v

dx dx= 4. 2d dv(cot v) cos ec v

dx dx=



5. d dv

(sec v) sec v tan vdx dx

= 6. d dv(cos ec v) cos ec v cot vdx dx

=

1. 2

d 1 dv(arcsin v)

dx dx1 v= 2. 2

d 1 dv(arccos v)

dx dx1 v=

3. 2d 1 dv

(arctan v)dx 1 v dx

= + 4. 2d 1 dv

(arc cot v)dx 1 v dx

= + 5.

2

d 1 dv(arc sec v)

dx dxv v 1= 6. 2

d 1 dv(arccos ec v)

dx dxv v 1=

(Composite Function) (Chain Rule) f x g f(x) gf x (gf)(x) = g(f(x))f(x) u = f(x) y = (gf)(x) y = g(f(x)) = g(u)

dydx

= g(f(x))f(x) = ( ) ( )d dg u u

du dx

u = f(x), y = g(u) = g(f(x)) dy du

,du dx

dydx

= dy dudu dx

1 f(x) = (2x 1)5 u = 2x 1 y = f(x) = (2x 1)5 = u5

dydx

= dy dudu dx

= ( ) ( )5d du 2x 1

du dx

= (5u4)(2) = 10u4

u f(x) = 10(2x 1)4



2 f(x) = 3 2

1

2x 1 u = 2x2 1

y = 3 2

1

2x 1 = 31

u =

13u

dydx

= dy dudu dx

= ( )1 23d du 2x 1du dx

= ( )431 u 4x3

u f(x) = ( ) ( )42 31 2x 1 4x3

= ( )4234x

3 2x 1

3 f(x) = 3 2

1

2x 1 u = 2x2 1

y = 3 2

1

2x 1 = 31

u =

13u

dydx

= dy dudu dx

= ( )1 23d du 2x 1du dx

= ( )431 u 4x3

u f(x) = ( ) ( )42 31 2x 1 4x3

= ( )4234x

3 2x 1



4 y = u2 u u = 3x2 dydx

dydx

= dy dudu dx

= ( ) ( )2 2d du u 3x

du dx

= ( ) ( )2u 1 6x u f(x) = ( ) ( )26x 1 6x = 236x 6x 5 12 5

168

x y ADC DEB+ +

y x12+

= x5

5y + 5x = 12x

x = 5

y7

168 dydt

dxdt

dxdt

= dx dydy dt

= ( )d 5 y 168dy 7

= 5

1687 = 120

120

12 ft

y x 5 ft

A B

C D

E



6 2 3 r , A A = r2

dr2

dt= r = 3

dAdt

dAdt

= dA drdr dt

=

dr2 r

dt

r 3

dAdt =

= 6 2 = 12 12

5

1) 1. 6y (2x 3)= + 2.

99x

y 13

=

3. 10

3F(t) 2

t

= +

4. ( ) 3 / 22g(t) 1 2t = 5. 2y 1 3x= 6.

4x 2

f(x) 13

= +

7. 3 2

1y

x 2x 3= +

8. 6

2x 1y

1 2x+ =

2) f(x) = 2x

x 1+ g(x) = 3x 1 F(x) F(x) = f(g(x)) 3) f(x) g(x) g(x) ( ) ( )2 3d f g x 3x f x 1

dx = +

4) 12 8 16



5) 0.2 1.5

(Implicit Function)

1. 2 2x y y+ =

2. x y x

2 2d d

(x y ) (y)dx dx

+ =

dx dy dy2x 2y

dx dx dx+ =

3. dydx

dy dy

2y 2xdx dx

=

4. dydx

dy

(2y 1) 2xdx

=

5. dydx

dy 2xdx 2y 1

= 1 xy + x 2y 1 = 10

( )d xy x 2y 1dx

+ = ( )d 10dx

( ) ( ) ( )d d d d dx y y x x 2 y 1dx dx dx dx dx

+ + = ( )d

10dx

dy dyx y 1 2dx dx

+ + = 0 dy dy

x 2dx dx

= - 1 - y

( )dy x 2dx

= - 1 - y dydx

= 1 y

x 2



2 x2 + 2xy + y2 + x = 5 (1, 1)

dy dy

2x 2x 2y 2y 1dx dx

+ + + + = 0

dy dy2x 2y

dx dx+ = - 2x 2y - 1

( )dy 2x 2ydx

+ = - 2x 2y - 1 dydx

= 2x 2y 12x 2y

+

(1, 1) x 1,y 1

dydx = =

= 2 2 12 2

+ =

54

6

1) 1. xy x 2y 1 + = 2. 3 3x y 1+ = 3. 2 3x xy y+ = 4. 3 5x y xy 2+ = 5. 2 3x y 2x y= 6. ( )22x 4 y 1 4+ =

2) 1 2 2x 3xy y 5+ + = (1, 1) 2 2 2x y 25+ = x = 3 3 xy 2= (-1, -2) 4 2 2x y 2x 3y 13 0+ + + = (1, 2)

f x f(x) f

f(x) f f f f f

f n n



y = 3x4 + 2x2

dydx

= 12x3 + 4x

dy dydx dx

= 36x2 + 4

dy dy dydx dx dx

= 72x + 4

2

2

dy dy d ydx dx dx

= 2 3

2 3

dy d y d ydx dx dx

=

y = f(x)

dy

y f (x)dx

= = , 2

2

d yy f (x)

dx = =

3

3

d yy f (x)

dx = = , ...,

n(n) (n)

n

d yy f (x)

dx= =

y = 3x4 + 2x2, y = 12x3 + 4x, y = 36x2, y = 72x, y(4) = 72

7 1)

1. 3 2f(x) 2x 3x 7= 2. 2f(x) 5x x 4= + 3. 3 2f(x) x 2x x 1= + 4. 4

1f(x) 100

x= +

5. 3 22 1

f(x) x 2x 2x

= + + + 2) x3 + 3xy y3 = 0



. . .6 2549

. 1 . . 2543 . . : . 2 6: , 2546

. .. : . Quota 2530 2548. IBC, :2546 . . Entrance . IBC, :2547

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