Explain why the expression x-(x+1)ln(x+1)

We'll assign a function f(x)=x-(x+1)ln(x+1) and we'll have to
prove that f(x)<0

To prove that the function has negative
values, for any positive values of x, we'll must show that it's derivative is also
negative.

We'll calculate it's derivative, using product
rule:

f'(x) =1 - (x+1)' * ln(x+1) –
(x+1)*[ln(x+1)]'

f'(x) = 1 - ln(x+1) -
[(x+1)/(x+1)]*1

f'(x) = 1 - ln(x+1) -
1

We'll eliminate like terms:

f'(x) =
-ln(x+1) < 0

The 1st derivative is
negative.

Since x is positive => the function is negative,
too.

Therefore, the inequality x-(x+1)ln(x+1)<0
is verified.