Example 9 Let $f(x)=\frac{x^{2}+6 x-15}{-x^{2}+10 x-17}$, find $f^{(n)}(x)$.
Solve $f(x)=\frac{x^{2}+6 x-15}{-x^{2}+10 x-17}$ has three fixed points $1, 3, 5$, which clearly satisfy the conditions of Theorem 5, then take $x_{1}=3, x_{2}=1$, we have $\varphi(x)=\frac{x-3}{x-1}, g(x)=2 x^{2}(k=2)$; or from $\frac{x^{2}+6 x-15}{-x^{2}+10 x-17}=$ $\frac{3(x-1)^{2}-2(x-3)^{2}}{(x-1)^{2}-2(x-3)^{2}}$, which also satisfies the conditions of Theorem 7, we have $\varphi(x)=\frac{x-3}{x-1}, g(x)=2 x^{2}(p=2)$, at this time $\varphi^{-1}(x)=\frac{x-3}{x-1}, g^{(n)}(x)=2^{2^{n}-1} \cdot x^{2^{n}}$, hence
$f^{(n)}(x)=\varphi^{-1}\left(g^{(n)}(\varphi(x))\right)=\frac{2^{2^{n}-1}(x-3)^{2^{n}}-3(x-1)^{2^{n}}}{2^{2^{n}-1}(x-3)^{2^{n}}-(x-1)^{2^{n}}}$ is the solution.