1. (Differentiation of
Function)

2. 2.1
f
y = f(x)
f (x 0 x) f(x 0 )
f (x 0 ) = lim
x 0 x
f (x 0 ) f
x0 f x0 f x0

3. f x
dy d
f (x) , y , f (x)
dx dx
y = f(x) x = x0
dy d
f (x 0 ) , y x=x 0
, f (x)
dx x=x 0 dx x=x 0

4. .
y = f(x) = x .
f (x) y= x x=4
.
f (x+ x) f (x)
f (x) = lim
x 0 x
x+ x x
= lim
x 0 x
( x+ x x )( x+ x x)
= lim
x 0 x( x+ x x)
(x+ x) x
= lim
x 0 x( x+ x x)

5. 1
f (x) = lim
x 0 x( x+ x x)
1 1
=
x x 2 x
#
1
. x=4 f (4) =
4
y= x (4, 2)
1 x -4
y-2 =
4 4
x
y= 1
4

6. 2.2
dc
0 c
dx
1 1
dx
x
dx
2 nx n -1
dx n n
dx
x
3
d d
f x cf (x) c f (x)
dx dx
c

7. 5 u = f(x) v = g(x)
d du dv
1. dx
u+v +
dx dx
( d du dv )
2. dx u v
dx dx
( d dv du
)
3. dx uv u
dx
+v
dx
( du dv
)
v u
d u
4. dx v dx
v 2
dx v 0
( )

8. f(x) = 1 + x + 2x 2 f (x)
d
f (x) = (1 x 2x 2 )
dx
d dx d
1 2x 2
dx dx dx
1 2(2x)
1 4x
#

9. f(x) = 1 + x + 2x 2 f (x)
d
f (x) = (1 x 2x 2 )
dx
d dx d
1 2x 2
dx dx dx
1 2(2x)
1 4x
#

10. f(x) = (x 2 +7x)(1+x 4 ) f (x)
d 2
f (x) 1 = (x 7x)(1 4x 4 )
dx
d 6
(x 7x 5 x 2 7x)
dx
dx 6 d 5 d 2 d
7x x 7x
dx dx dx dx
5 dx 5 dx
6x 7 2x + 7
dx dx
6x 5 35x 4 2x +7

11. d 2
2f (x) = dx
(x 7x)(1 x 4 )
d d 2
(x 2 7x) (1 x 4 ) (1 x 4 ) (x 7x)
dx dx
(x 2 7x)(4x 3 ) (1 x 4 )(2x 7)
4x 5 28x 4 2x+2x 5 7x 4 7
6x 5 35x 4 2x+7 #

12. 4x+3
f(x) = f (x)
x2 1
d 4x+3
f (x) =
dx x 2 1
2 d d 2
(x 1) (4x+3) (4x+3) (x 1)
dx dx
(x 2 1) 2
(x 2 1)(4) (4x+3)(2x)
(x 2 1) 2
4x 2 4 8x 2 6x
(x 2 1) 2
4 6x 4x 2
(x 2 1) 2 #

13. 3 1
f(x) = x f (2)
3 x2
d 1
f (x) = x3
dx 3 x2
d 3 d 1
x
dx dx 3 x 2
(3 x 2 )(0) ( 2x)
3x 2
(3 x 2 ) 2
2x
3x 2
(3 x 2 )2
2 2(2)
f (2) = 3(2) 12 4 8 #
(3 4)2

14. f I
a a I a
1. f a f a
f a f a
2. ,
d n
6 n x 0
dx
x nx n -1

15. dy
y = 3x 4 12x 3
dx
d dy
36x 2
dx dx
d dy dy
72x
dx dx dx
y = f(x)
dy d2y
y = f (x) , y = f (x)
dx dx 2
3
dy dn y
3
y = f (x) , y (n) = f (n) (x)
dx dx n

16. y 4 3y 4x 3 5x 1 y
3 dy dy
4y 3 12x 2 5
dx dx
dy 12x 2 5
dx 4y3 3
d d
(4y3 3) (12x 2 5) (12x 2 5) (4y3 3)
d2 y dx dx
dx 2 (4y3 3) 2
3 2 2 dy
(4y 3)(24x) (12x 5)(12y )
dx
(4y3 3) 2

17. dy d2y
dx dx 2
3 2 2 12x 2 5
(4y 3)(24x) (12x 5)(12y )
2
d y 4y3 3
dx 2 (4y3 3) 2
d2 y 24x(4y3 3) 2 12y 2 (12x 2 5) 2
dx 2 (4y3 3)3 #

18. 2.3 (Chain Rule)
h = f g
h (x) (f  g) (x) f g(x) g (x)
y = f(u) u = g(x) y = h(x)
dy dy du
dx du dx

19. u = u(x) y = u n ,n
dy
dx
dy dy du
dx du dx
d n du
u
du dx
du
nu n 1
dx #

20. 7u = u(x) x y = un n
dy d n du
u nu n 1
dx dx dx
d 1 x =1 F (0) = 2
F x
dx x
1
u= x
x
x =1 u=0
dF dF du
dx du dx
d 1
F (u) x
dx x
1 1
F (u) 3
2 x 2
2x

21. dF 1 1
F (0) F (0) = 2 #
dx x 1 2 2
2 y = f  g  h (x)
y (x) = (f  g  h ) (x) = (f  (g  h)) (x) = f (g  h(x))(g  h) (x)
f (g  h(x)) g (h(x)) h (x)
u = g  h (x) , v = h(x)
dy dy du dv
= f (u) g (v) v (x)
dx du dv dx

22. 1+ u 1 2 dy
y= , u= , v=x
1 u v dx
dy d 1 u d 1 d 2
(x )
dx du 1 u dv v dx
(1 u) (1 u)(-1) 1
(2x)
(1 u) 2 v 2
2 2x 4x
(1 u)2 v2 v 2 (1 u) 2 #

23. 2.4
x
y y
y x
dx
x
dy
y x y

24. dy
dx
3x 5 3x 4 y3 4y 2 0
(x y) 2 (x y) 2 3x 5 xy 6 x =1, y = 1
. x
d d
(3x 5 3x 4 y3 4y 2 ) (0)
dx dx
d 5 3 d d 3 d 2
3 x 3(y x4 x4
y ) 4 y 0
dx dx dx dx
4 3 3 dy
4 2 dy
15x 3(4x y 3x y ) 8y 0
dx dx
dy 12x 3 y3 15x 4
dx 8y 9x 4 y 2 #

25. d d d
(x y) 2 (x y) 2 (3x 5 xy 6 )
dx dx dx
d d dy
2(x y) (x y) 2(x y) (x y) 15x 4 y 6 6xy5
dx dx dx
dy dy dy
2(x y)(1 ) 2(x y)(1 ) 6xy5 15x 4 y6
dx dx dx
dy
(4x 6xy5 ) 15x 4 y6 4y
dx
dy 15x 4 y6 4y
dx 4x 6xy5
dy
x =1 , y = 1 6 #
dx

26. 2.5
1-1
1
f f
1 1
f f
1
y f x x f y
1
y f f y
y
1 1 1 dy dx
1 f f y f y f x f y
dx dy
dx 1
dy dy
dx

27. 3 2 dy
x y 1
dx
1
x y2 1 3
2
dx 1 2
y 1 3 2y
dy 3
2y
2
3 y2 1 3
2
dy 3 y2 1 3
#
dx 2y

28. 2.6
(Differentials)
dx, dy
y f x x
x
x dx dx x
y f dy df
dy df f x dx

29. y
(1)
y x 2 3x 5
dy d 2
x 3x 5 2x 3
dx dx
dy 2x 3 dx
#
(2) y (2)
x2 4
1
dy d 2 d 2 x
x 4 x 4 2
dx dx dx x2 4
x
dy dx
2
x 4
#

30. 2.7
f
f a,b
1. f a, b
2. f a f b 0
3. c a, b f c 0

31. (The Mean - Value
Theorem) f
f a,b
f
1. a, b
f a f b
c a, b f c
2. b a

32. c 1,2
f x x2
f x x2 1, 2
f a f b
f c
b a
f 2 f 1 22 11
3
2 1 1
f x 2x f c 2c
2c 3
3
c
2

33. 40
f x x
1
f x 1
2
2x
a 36 b 40 , c 36
f 40 f 36
f 36
40 36
1 40 36
2 36 4
4
40 36
2 36
6.33 #