Example 5 If $f(x)=\sqrt{19 x^{2}+93}$, find $f^{(n)}(x)$.
Solution: First, find the fixed point of $f(x)$. From $\sqrt{19 x^{2}+93}=x$, we get $x^{2}=-\frac{31}{6}$. Therefore,
$$
\begin{array}{l}
f(x)=\sqrt{19\left(x^{2}+\frac{31}{6}\right)-\frac{31}{6}}, \\
f^{(2)}(x)=\sqrt{19^{2}\left(x^{2}+\frac{31}{6}\right)-\frac{31}{6}}, \\
\ldots \ldots
\end{array}
$$

By induction, we get $f^{(n)}(x)=$
$$
\sqrt{19^{n}\left(x^{2}+\frac{31}{6}\right)-\frac{31}{6}} .
$$