Example 2 Let $f(x)=\frac{x}{\sqrt{1+x^{2}}}$, find $f^{(n)}(x)$
Solve $f^{2 x \prime}(x)=f(f(x))=\frac{x}{\sqrt{1+2 x^{2}}}, f^{(3)}(x)=f\left(f^{(2)}(x)\right)=\frac{x}{\sqrt{1+3 x^{2}}}, \cdots$ Thus, we conjecture that $f^{(n)}(x)=\frac{x}{\sqrt{1+n x^{2}}}$. It is very easy to prove this by mathematical induction. The proof is left to the reader.