Example 6 If $f(x)=2 x^{2}-1$, find $f^{(n)}(x)$.
Let $g(x)=2 x, \varphi(x)=\arccos x$,
then $\varphi^{-1}(x)=\cos x$.
$$
\begin{array}{l}
f(x)=2 x^{2}-1=2 \cos ^{2}(\arccos x)-1 \\
=\cos 2(\arccos x)=\varphi^{-1}(g(\varphi(x))) .
\end{array}
$$

So $f^{\varphi} \sim g$. And $g^{(n)}(x)=2^{n} x$, therefore
$$
\begin{array}{l}
f^{(n)}(x)=\varphi^{-1}\left(g^{(n)}(\varphi(x))\right) \\
=\cos \left(2^{n} \arccos x\right) .
\end{array}
$$