1109291212434690_1212270774412.pdf
TRANSCRIPT
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: 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)
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: Derivative of Function 2
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
= ( ) ( )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
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: Derivative of Function 3
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 4
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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)
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: Derivative of Function 5
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 6
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 7
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 8
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
= ( )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
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: Derivative of Function 9
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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)
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: Derivative of Function 10
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 11
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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) 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))
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: Derivative of Function 12
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
7 f g x (fg)(x) = f(x)g (x) + f(x)g(x) F(x) = f(x)g(x) F(x) = ( ) ( ) ( ) ( )
h 0
f x h g x h f x g xlim
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
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: Derivative of Function 13
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
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: Derivative of Function 14
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 15
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 16
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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+
-
: Derivative of Function 17
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 18
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
+= +
-
: Derivative of Function 19
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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=
-
: Derivative of Function 20
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 21
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 22
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 23
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 24
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 25
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 26
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
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
-
: Derivative of Function 27
-- For Educational Purpose only. Not for commerce bySupanut Chaidee ::[MoDErN_SnC]::
. . .6 2549
. 1 . . 2543 . . : . 2 6: , 2546
. .. : . Quota 2530 2548. IBC, :2546 . . Entrance . IBC, :2547