Clearly
dy
dt  = 
(
)
t t1
d
a
dt
+
 =
1
1
log
at t d
t
a
dt
t
+

+
⋅




=
1
12
1
log
at t
a
t
+ 

−



Similarly
dx
dt  =
1
1
1
a
d
a t
t
t
dt
t
−




+
⋅
+







=
1
2
1
1
1
a
a t
t
t
−




+
⋅
−







dx
dt  ≠ 0 only if t ≠ ± 1 Thus for t ≠ ± 1,
Rationalised 2023-24
 MATHEMATICS
144
dy
dy
dt
dx
dx
dt
=
 =
a
t
a
a t
t
t
t t
a
+
−
−


+

⋅
−


1
2
1
2
1
1
1
1
1
log
=
1
1
log
1
t t
a
a
a
a t
t
+
−

+



Example 43 Differentiate sin2 x w r t