f (x) = cos^–1 (2x² – 1) = cos^–1(cos 2θ)

$=\{\begin{array}{l}2\mathrm{\theta };\mathrm{\theta }\in {[0,\frac{\mathrm{\pi }}{2}]}_{}\Rightarrow \mathrm{x}\in [0,1]\\ 2\mathrm{\pi }-2\mathrm{\theta };\mathrm{\theta }\in {(\frac{\mathrm{\pi }}{2},\mathrm{\pi }]}^{}\Rightarrow \mathrm{x}\in [-1,0)\end{array}$

 $\mathrm{g}\left(\mathrm{x}\right)=\sqrt{1-{\mathrm{x}}^2}=\sqrt{1-{\cos }^2\mathrm{\theta }}=\mathrm{sin\theta }$

 $\text{At\hspace{0.17em}\hspace{0.17em}x }=\frac{3}{5}$

 $\frac{\mathrm{d}\left(\mathrm{f}\left(\mathrm{x}\right)\right)}{\mathrm{d}\left(\mathrm{g}\left(\mathrm{x}\right)\right)}=\frac{2}{\mathrm{cos\theta }}=\frac{2}{3/5}=\frac{10}{3}$

 $\mathrm{At}\mathrm{x}=\frac{-3}{5}$

 $\frac{\mathrm{d}\left(\mathrm{f}\left(\mathrm{x}\right)\right)}{\mathrm{d}\left(\mathrm{g}\left(\mathrm{x}\right)\right)}=\frac{-2}{\mathrm{cos\theta }}=\frac{-2}{-3/5}=\frac{10}{3}.$