Show that `y=log(1+x)-(2x)/(2+x),\ x >-1` is an increasing function of `x` throughout its domain.

Explanation

`f(x) = log (1+x) - (2x)/(2+x)`
` rArr f'(x) = 4/(1+x) - 4/(2+x)^(2)`
` rArr f'(x) = ((2+x)^(2)-4(1+x))/((1+x)(2+x)^(2))`
`= (x^(2))/((1+x)(2+x)^(2))`
Now, f(x) is increasing,
` rArr f'(x) ge 0`
`rArr (x^(2))/((1+x)(2+x)^(2)) ge 0`
`rArr (1+x) gt 0`
` rArr x gt - 1`
`:." In " x gt -1, f(x) ` is increasing.