$x(a^{1/x}-1)$ is decreasing 
Prove that $f(x)=x(a^{1/x}-1)$ is decreasing on the positive $x$ axis for $a\geq 0$.

My Try:
I wanted to prove the first derivative is negative.
$\displaystyle f'(x)=-\frac{1}{x}a^{1/x}\ln a+a^{1/x}-1$. But it was very difficult to show this is negative. Any suggestion please.

Using simple algebra it is possible to prove that $g(x) = f(1/x) = \dfrac{a^{x} - 1}{x}$ is strictly increasing for $x > 0, a > 0, a \neq 1$ and $x$ being rational. The extension to irrational values of $x$ is easily done by considering sequences of rationals converging to $x$.
