Here we have to find the interval in which f(x) is increasing. Let's first calculate f'(x) As we know that
d
dx
{f(x)±g(x)}=
d{f(x)}
dx
±
d{g(x)}
dx
,
d(logx)
dx
=
1
x
,for x>0 and
d
dx
[
f(x)
g(x)
]=
g(x).
d(f(x))
dx
−f(x).
d(g(x))
dx
[g(x)]2
⇒f′(x)={
1
1+x
−
(1+x).1−x.1
(1+x2)
}=
x
(1+x)2
As we know that for an increasing function say f(x) we have f′(x)≥0 ⇒x≥0.........(∵(1+x)2≥0) Hence, the given function is increasing for the interval [0, ∞)