Proving that $f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$ is increasing for $x \in [1, 2]$

I want to show that the function $f: [1, \infty) \rightarrow \mathbb{R}$ given by $$f(x) = (2^x-1)^{1/x}+ (2^x+3^x-1)^{1/x}$$ is increasing for $x \in [1, 2]$. By plotting it this seems to be true, however I have a hard time working with the derivative.