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

$\frac{2}{x^2(\log 5)(\log x^2)}$ $\frac{2}{x(\log 5)(\log x)}$ $\frac{1}{x(\log 5)(\log x^2)}$ $\frac{1}{x(\log 5)(\log x)}$

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

$\log_5\log x^2=\frac{\log(\log x^2)}{\log_5}$ differentiating w.r.t x we get $\frac{1}{\log_5\log x^2}\frac{d}{dx}(\log x^2)$ $=\frac{1}{x^2\log_5\log x^2}\frac{d(x^2)}{dx}=\frac{2x}{x^2\log_5\log x^2}$ $=\frac{2}{x\log 5\log x^2}=\frac{2}{x\log 5\log x}$ $=\frac{1}{x(\log 5)(\log x)}$