Differentiation of log_5 (log x^2) w.r.t.x is

$\log_5\log x^2=\frac{\log(\log x^2)}{\log_5}$

differentiating w.r.t x

we get $\frac{1}{\log_5\log x^2}\frac{d}{dx}(\log x^2)$

$=\frac{1}{x^2\log_5\log x^2}\frac{d(x^2)}{dx}=\frac{2x}{x^2\log_5\log x^2}$

$=\frac{2}{x\log 5\log x^2}=\frac{2}{x\log 5\log x}$

$=\frac{1}{x(\log 5)(\log x)}$