2012 differentiationtutorial solutions barely passed
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CJC MATHEMATICS DEPARTMENT
2012 JC1 H2 MATHEMATICS
TOPIC: DIFFERENTIATION
QUESTIONS
1. [N00/I/4] 3)(f xx =
x
xxxx
x δ
δ
δ
33
0
)(lim)('f
−+=
→
( ) ( )x
xxxxxxx
x δ
δδδ
δ
33223
0
33lim
−+++=
→
( ) ( )( )22
033lim xxxx
xδδ
δ++=
→
23x= (shown)
____________________________________________
2. [2007/MJC/I/9]
___________________________________________
Differentiation of Standard Functions
3(a) ( ) xxxxxx
sincossind
d+=
(b) ( )( ) ( ) ( ) ( ) 22tan2sec2sec32secd
d 23 ⋅= xxxxx
( ) ( )xx 2tan23sec62
=
(c) ( ) ( )xxxxx
212
11cos1sin
d
d2
1222 −
+⋅+=
+
1
1cos
2
2
+
+=
x
xx
____________________________________________
Differentiation of Exponential and Logarithmic
Functions
4(a) ( )
=
10log
5log
d
d5log
d
d10
e
e x
xx
x
x
e
ex
x
x
x
x
10
10
10
log
log
10log
1or
10ln
1
5
5
10ln
1
10ln
5ln
d
d
==
=
=
(b) ( )( ) ( )
x
xx
xx cos1
sincos12
1
cos1lnd
d2
1
+
−+
=+
−
( )x
x
cos12
sin
+
−=
(c) ( )( )[ ]( )32lnd
d 23 ++ xxx
( )( ) ( )( )
( )( )32
332223
223
++
+++=
xx
xxxx
( )( )( )
( )( )( )32
495
32
9342
23
3
23
33
++
++=
++
+++=
xx
xxx
xx
xxxx
____________________________________________
Implicit Differentiation
5. xyxy =+ cossin
( )
xy
xy
x
y
xyxyx
y
yx
yxx
x
yy
−
+=
+=−
+=−
cos
sin
d
d
sincosd
d
d
dsin
d
dcos
____________________________________________
Logarithmic Differentiation
6 Let xxy =
( )
(shown) 2
ln2
d
d
ln2
11
d
d1
2
1ln
d
d1
lnln
2
1
+=
+=
⋅+=
=
−
x
xx
x
y
xxxx
y
y
xxx
x
x
y
y
xxy
x
____________________________________________
( )xy 'f=
y
O 2a 4a x
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Differentiation of Inverse Trigonometric Functions
7. [2010/CJC Promo/1/5]
1tan 21 −= −xy
−
−+=
−
)2()1(2
1
)11(
1
d
d2
1
2
2xx
xx
y
1
1
2 −=
xx
___________________________________________
Parametric Differentiation
8. [2008/TJC/Promo/5i]
21
1
tx
+= 13 += ty
( ) ( )ttt
x21
2
1
d
d2
32 −
+−= 23d
dt
t
y=
( )2
321 t
t
+
−=
( ) ( )2
32
3
22
2 131
3d
d
d
d
d
dtt
t
tt
x
t
t
y
x
y+−=
+⋅−=⋅=