MATH1054 Worksheets with Solutions

,8 3 8
. . Exercise du = 4x3 + 2x2 + 4x2 + 2x + 2x + 1

dx = 4x3 + 6x2 + 4x + 1




v = (x3 + 2x2 + 1)4
y = 29 2 = 5x + 3 W = x3 + 2x2 + 1 V = 04

928 d &w 3x2 4x dr 40
dy
= = = + =

di di dw

dr = dr x dw
di dw dx
dy dy x
=



= 4(23 + 2x2 + 1)3(3x2 + 4x) -
= 928x5
=
4528 =
45(5x + 38/ ndv + vdu
&x di

= (x2 + x + 1)24(x3 + 2x2 + 1)3(3x2 + 4x)
+ (x3 +1)4 2x2 +
20 - z 4x3 2x +
1 (2x2 + 2x + 1)(2x + 1)
Y = = -




625 1 (x2 1)24(23 2x2 + 1)3(3x2 + 4x)
dy d
= = + x + +
=


+ (x3 + 2x2 + 1)4
=
62 (12x2 2) - = 1225(6x2- 1) - 2(x2 + x + 1)(2x + 1)
=
(x2 +x+ 1)2(x3 + 2x2 1)3(12x2 16x)
+ + +


1)+ 2(x2 + x + 1)(2x + 1)
dy =dy the
Xd (n3 + 2x2+



=
12 (4x3 -
2x + 1)(6x 2
- 1)



*

u = (x2 + x + 1)

=duxd
u = z2 - 2 = x2 + x + 1 -


=
2 dz = 2x + 1

dx

z(x2 + 1)(2x + 1) -
dr en =
x +


2x + 1)(2x + 1)
-
= (2x2 +

, 2(4 + x2)i v = (4 + x2)
u = x
w = 4+ x2 v =wi
=w
ar
du =
dw = 2x

dr
did


dr =drX
dr =
tw e



=
-
w x


(4 + x2) Ex
-




=



I R


(4 + x2

u dv + vdu
dx di


=
x R + (4 + x2)
(4 + x2
= x2 (4 + x2 xJ4+ 22
+




(4 + x2 1 J
= x2 + (4 + x2)
(4 + x2

= 4+ 2x2 = z(z + x2)
14 + x2 (4 + x2

, (x2 + 1)
z = x3 + 1


n =
25

du = 17 dz = 2x

d2 3 dx

dy = duxdz


on
de (2)

= 25
3

1)
2(x2
=
+




192
= 24

3(x2+ 1) - 3 3 + 12




↳ Expand
2)(j)
into x + 1
+
R




1
= x
+
+ 2



2
f'(x)
-



= 1 -
1x

=
1-1
x