B-4. $p(x)$ is a non-zero polynomial of degree less than 1992, and $p(x)$ has no common factor with $x^{3}-x$. Let
$$
\frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^{3}-x}\right)=\frac{f(x)}{g(x)},
$$
where $f(x), g(x)$ are polynomials. Find the smallest possible degree of $f(x)$.
The minimum possible degree of $f(x)$ is 3984.
By the division algorithm, we let $p(x) = (x^3 - x)q(x) + r(x)$, where $q(x)$ and $r(x)$ are polynomials, and the degree of $r(x)$ is less than 3, and the degree of $q(x)$ is less than 1989. Then,
$$
\cdot \frac{d^{1992}}{d x^{1992}}\left(\frac{p(x)}{x^3 - x}\right) = \frac{d^{1992}}{d x^{1992}}\left(\frac{r(x)}{x^3 - x}\right).
$$

Since $p(x)$ and $x^3 - x$ have no common factors, $r(x)$ and $x^3 - x$ also have no common factors.

Let $p_0(x)$ be a polynomial that is coprime with $x^3 - x$ and has degree $n < 3$. We will use induction to show that
$$
\frac{d^k}{d x^k}\left(\frac{p_0(x)}{x^3 - x}\right) = \frac{p_k(x)}{(x^3 - x)^{k+1}},
$$

where $p_k(x)$ is a polynomial that is coprime with $x^3 - x$ and $\operatorname{deg}(p_k) = \operatorname{deg}(p_0) + 2k < 3(k+1)$. We have
$$
\begin{array}{l}
\frac{d}{d x}\left(\frac{p_k(x)}{(x^3 - x)^{k+1}}\right) \\
= \frac{(x^3 - x)p_k'(x) - (k+1)(3x^2 - 1)p_k(x)}{(x^3 - x)^{k+2}} \\
= \frac{p_{k+1}(x)}{(x^3 - x)^{k+2}}
\end{array}
$$

The polynomial $p_{k+1}(x)$ is coprime with $x^3 - x$, because the left term is divisible by $x^3 - x$ and is coprime with the right term (the left term refers to $(x^3 - x)p_k'(x)$, and the right term refers to $(k+1)(3x^2 - 1)p_k(x)$ — translator's note). If $p_k(x) = ax^n +$ lower order terms, $a \neq 0$, then the highest degree coefficient of $p_{k+1}(x)$ is $a(n - 3(k+1))$, and since $n \neq 3(k+1)$,
$$
\begin{aligned}
\operatorname{deg}(p_{k+1}) & = \operatorname{deg}(p_k) + 2 = \operatorname{deg}(p_0) + 2k + 2 \\
& < 3 + 3k + 2 < 3(k+2).
\end{aligned}
$$

This completes our induction.
Thus, $\frac{d^{1992}}{d x^{1992}}\left(\frac{r(x)}{x^3 - x}\right) = \frac{f(x)}{g(x)}$,

where $\operatorname{deg}(f) = \operatorname{deg}(r) + 2(1992) \geq 3984$. The minimum value is achieved only when $r(x)$ is a constant.
($\operatorname{deg}(p)$ denotes the highest degree of the polynomial $p(x)$ — translator's note)