Find \(n^{th}\) derivative of \(\frac{x^2}{(x-1)(x-2)(x-3)}\). [The numerator is faint in the source scan; it appears to be \(x^2\).]

(Solved taking the numerator as \(x^2\), as it appears in the scan.)

Partial fractions: \(\frac{x^2}{(x-1)(x-2)(x-3)}=\frac A{x-1}+\frac B{x-2}+\frac C{x-3}\).

\(A=\frac{1}{(-1)(-2)}=\frac12,\quad B=\frac{4}{(1)(-1)}=-4,\quad C=\frac{9}{(2)(1)}=\frac92\) (check at \(x=0\): \(-\frac12+2-\frac32=0\) ✓).

\(y=\frac12\cdot\frac1{x-1}-\frac4{x-2}+\frac92\cdot\frac1{x-3}\)

Using \(D^n\left(\frac1{x-a}\right)=\frac{(-1)^nn!}{(x-a)^{n+1}}\):

\(y_n=(-1)^n\,n!\left[\frac1{2(x-1)^{n+1}}-\frac4{(x-2)^{n+1}}+\frac9{2(x-3)^{n+1}}\right]\)