Is it possible to show that some function $f(x)$ is larger than $g(x)$ if I can show that its derivative is larger for some $x>x_0$?
As an example I can think of $f(x)=(1+x)^n , \ g(x)=1+nx+ \frac{n(n-1)x^2}{2}$, which derivatives are $n(1+x)^{n-1}$ and $n+n(n-1)x$ and the use Bernoulli inequality to show that f'(x)>g'(x) for $x>0$.