Example 5 Let $f(x)=2 x^{2}-1$, find $f^{(n)}(x)$
Let $g(x)=2 x, \varphi(x)=\arccos x$, then $\varphi^{-1}(x)=\cos x$. And $f(x)=\varphi^{-1}(g(\varphi(x)))$ Thus, $f^{(m)}(x)=\varphi^{-1}\left(g^{(n)}(\varphi(x))\right)=\cos \left(2^{n} \arccos x\right)$, this iterative result is the Chebyshev polynomial.