Example 6 Given $x_{n+1}=\frac{x_{n}}{\left(1+c x_{n}^{k}\right)^{\frac{1}{k}}}$, where $k \in$ $\mathbf{N}, c \in \mathbf{R}, c \neq 0$, and $x_{n}>0$. Find $x_{n}$.
Let $y_{n}=\frac{1}{x_{n}^{k}}$, then $x_{n}=\left(\frac{1}{y_{n}}\right)^{\frac{1}{k}}$. The original expression becomes $y_{n+1}=y_{n}+c$. Therefore, $y_{n}=y_{1}+(n-1) c$. Thus, we have
$$
x_{n}=\frac{x_{1}}{\sqrt[k]{1+(n-1) c x_{1}^{k}}} .
$$