Question

Find the n^th derivative of `x^3/((x+1)(x-2))`

Answer 1

Let y= `x^3/((x+1)(x-2))=x^3/(x^2-x-2)`

∴` y=x+1 (3x+2)/(x^2-x-2)` 

∴` y=x+1+(3x+2)/((x+1)(x-2))`

∴ `y=x+1+(1/3)/(x+1)+(8/3)/(x-2)`    (By Partial Fraction) 

Taking `n^(th)` order derivative, `y_n`=`0+0+1/3. (n!.1^n(-1)^n)/((x+1)^n+1)+8/3. (n!.1^n(-1)^n_)/((x-2)^n+1)`

`{"If" = 1/(ax+b)"then"   y_n= (n!a_n(-1)^n)/((ax+b)^n+1)}`

∴` (y_n=n!(-1)^n)/3[1/((x+1)^(n+1))+8/((x-2)^n+1)]`