Can the inequality $\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} \geq 6$ be proved with differentiation? $$\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} \geq 6,\quad \text{with}\quad a,b,c >  0$$
I could do it with letting $x=\frac{a}{b}$, $y=\frac{b}{c}$, $z=\frac{c}{a}$, but I wonder if it is solvable somehow with differentiation.

By AM-GM for positive variables:
$$\sum_{cyc}\frac{b+c}{a}\geq2\sum_{cyc}\frac{\sqrt{bc}}{a}\geq6\sqrt[3]{\prod_{cyc}\frac{\sqrt{bc}}{a}}=6.$$
AM-GM it's the Jensen inequality for $f(x)=\ln{x}$ because $f''(x)=-\frac{1}{x^2}<0$.
