Example 11 Given $x>y>0$, and $x y=1$. Find the minimum value of $\frac{3 x^{3}+125 y^{3}}{x-y}$.
Let the minimum value (target) be a positive number $t$, then
$$
\frac{3 x^{3}+125 y^{3}}{x-y} \geqslant t \text {, }
$$

which means $3 x^{3}+125 y^{3}+y t \geqslant x t$.
By $x y=1$ and the AM-GM inequality, we have
$3 x^{3}+125 y^{3}+y t$
$=x^{3}+x^{3}+x^{3}+125 y^{3}+y t$
$\geqslant 5 \sqrt[5]{x^{9} \cdot 125 y^{4} t}=5 \sqrt[5]{125 x^{5} t}$
$=5 x \sqrt[5]{125 t}$.
Let $5 x \sqrt[5]{125 t}=x t$, solving for $t$ gives $t=25$.
Note that the equality in the above inequality holds if and only if $x^{3}=125 y^{3}=y t$.

Therefore, when $x=\sqrt{5}, y=\frac{\sqrt{5}}{5}$, $\frac{3 x^{3}+125 y^{3}}{x-y}$ achieves its minimum value of 25.