Question

Find nth derivative of `1/(x^2+a^2.`

Answer 1

y = `1/(x^2+a^2`

`y = 1/((x+ai)(x-ai))`

Let `1/((x+ai)(x-ai))="A"/(x+ai)+B/(x-ai)`

`1 = "A"(x-ai)+"B"(x+ai)`

Put x = ai

1 = B(2ai)

B = `1/(2ai)`

Put x = -ai, we get

`"A"=-1/(2ai)`

`therefore y=(1/(-2ai))/(x+ai)+(1/(2ai))/(x-ai)`

`therefore y = 1/(2ai)[1/(x+ai)-1/(x-ai)]`

Diff n times, we get

`y_n=1/(2ai)[((-1)^n n!)/((x-ai)^(n+1))-((-1)^n n!)/(x+ai)^(n+1)].`