By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Let f be a function defined on the positive real axis with values in R, such that f(x) = x - ln(1+x). Differentiating this function, we obtain, f'(x) = (x - ln(1+x))' = x' - ln'(1+x) = 1 - 1/(1+x). Since, x > 0, we have 1 + x > 1, and so 1/(1+x) < 1. So, 1 - 1/(1+ x) > 0. So, f'(x) > 0, for all positive x. So, by L'Hospital Rule, f(x) is strictly increasing. Thus, f(x) > lim f(x) when x -> 0 = 0 - ln (1+0) = 0. So, f(x) > 0, for all positive. x.