Calculating the $n^\text{th}$ derivative How do we calculate the $n^{\text{th}}$ derivative for
$$
\frac{x^3}{(x-a)(x-b)(x-c)}?
$$
How can I obtain the partial fraction for the given term?

If we assume that $a,b,c$ are distinct numbers, from:
$$ \text{Res}\left(\frac{z^3}{(z-a)(z-b)(z-c)},z=a\right)=\frac{a^3}{(a-b)(a-c)}\tag{1}$$
it follows that:
$$\begin{eqnarray*} f(z)&=&\frac{z^3}{(z-a)(z-b)(z-c)}\tag{2}\\&=&1+\frac{a^3}{(a-b)(a-c)(z-a)}+\frac{b^3}{(b-a)(b-c)(z-b)}+\frac{c^3}{(c-a)(c-b)(z-c)}\end{eqnarray*}$$
and we are able to compute the derivatives of $f(z)$ from the derivatives of $\frac{1}{(z-a)}$, that are straightforward to compute.
