t x, the following function:
(i)
12
3
2
2
4
x
x
+
+
+
(ii)    log7 (log x)
Solution
(i)
Let y = 
12
3
2
2
4
x
x
+
+
+
= 
1
1
2
2
2
(3
2)
(2
4)
x
x
−
+
+
+
Note that this function is defined at all real numbers 
x > −32 Therefore
dy
dx  =
1
1
1
1
2
2
2
2
1
1
(3
2)
(3
2)
(2
4)
(2
4)
2
2
d
d
x
x
x
x
dx
dx
−
− −


+
⋅
+
+ −
+
⋅
+




= 1
2 3
2
3
21
2
4
4
21
2
23
(
)
( )
(
)
x
x
x
+
⋅
− 
 

+
⋅
−
−
=
(
)
3
2
2
3
2
2 3
2
2
4
x
x
x
−
+
+
This is defined for all real numbers 
x > −32 Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
141
(ii)
Let y = log7 (log x) = log (log )
log7
x  (by change of base formula)