p.t the value of nth derivative of x^3/(x^2-1) for x=0 is 0 if n is even & (-n!) if n is odd

Question

its a maths ques

Answer 1

Let y=x³/(x²-1)=x+x/(x²-1).

Let x/(x²-1)=A/(x-1)+B/(x+1), then Ax+A+Bx-B=x; A-B=0 and A+B=1, so A=B=½.

y=x+½(x-1)^-1+½(x-1)^-1); y'=y⁽¹⁾=1-½(x-1)^-2-½(x+1)^-2; y⁽²⁾=½[2(x-1)^-3+2(x+1)³]; etc.

y⁽ⁿ⁾=½(-1)ⁿ[n!(x-1)^-(n+1)+n!(x+1)^-(n+1)]. But y'=1-½-½=0.

When x=0, y⁽ⁿ⁾=½(-1)ⁿ[n!(-1)^-n-1+n!(1)^-n-1].

When n>1 is odd: y⁽ⁿ⁾=-½[n!+n!]=-n!; when n is even: y⁽ⁿ⁾=½[-n!+n!]=0 QED