, all x such that 2x + 1 ≤ 1 + 4x We
may rewrite this as 2 ≤ 1
2x  + 2x which is true for all x Hence the function
is defined at every real number By putting 2x = tan θ, this function may be
rewritten as
f(x) =
1
1
2
sin
1
4
x
x
+
− 



+

= sin−
⋅
+ (
)






1
2
2
2
1
2
x
x
=
1
2tan2
sin
1
tan
−
θ




+
θ


= sin –1 [sin 2θ]
= 2θ = 2 tan – 1 (2x)
Thus
f ′(x) =
(
)
2
1
2
(2 )
1
2
x
x
dxd
⋅
⋅
+
=
2
(2 )log2
1
4
x
x ⋅
+
=
1
2
log2
1
4
x
x
+
+
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
143
Example 41 Find  f ′(x) if f(x) = (sin x)sin x for all 0 < x < π