Question

find the nth derivatives of 1/x^4 - a^4​

Answer 1

Given :  1/(x⁴ - a⁴)

To Find : nth derivative

Solution:

 1/(x⁴ - a⁴)

= 1/(x² + a²)(x² - a²)

1/(x² + a²)(x² - a²) = A/(x² + a²)  + B/(x² - a²)

=> 1 = A(x² - a²) + B(x² + a²)

x² = -a²

=> A  = -1/2a²

x² = a²

=> B = 1/2a²

= (-1/2a²)1/(x² + a²)  + (1/2a²)1/(x² - a²)

(1/2a²)1/(x² - a²)  =  (1/2a²)1/(x + a)(x - a)

(1/2a²)1/(x + a)(x - a)   = A/(x + a)  + B/(x - a)

=>  (1/2a²) = A(x - a)  + B(x + a)

x = - a

=> A = -1/2a³

x = a

=> B = 1/2a³

(1/2a²)1/(x² - a²)   =  (-1/2a³)1/(x + a)  +  (1/2a³)1/(x - a)

(-1/2a²)1/(x² + a²)  =  (-1/2a²)1/(x + ia)(x - ia)

=>  (-1/2a²)1/(x + ai)(x - ai)   = A/(x + ai)  + B/(x - ai)

=>  (-1/2a²) = A(x - ai)  + B(x + ai)

x = - ai

=> A =  1/2a³i   = - i/2a³

x = ai

=> B = -1/2a³i   = i/2a³

(-1/2a²)1/(x² + a²) =   (-i/2a³)1/(x + ai)  +  (i/2a³)1/(x - ai)

 1/(x⁴ - a⁴)  =  (-1/2a³)1/(x + a)  +  (1/2a³)1/(x - a) +  (-i/2a³)1/(x + ai)  +  (i/2a³)1/(x - ai)

nth derivative of 1/(x + a)   = (-1)ⁿn!/(x + a)ⁿ

Hence nth derivative

(-1/2a³)(-1)ⁿn!/(x + a)ⁿ  +  (1/2a³)(-1)ⁿn!/(x - a)ⁿ +  (-i/2a³)(-1)ⁿn!/(x + ai)ⁿ  +  (i/2a³)(-1)ⁿn!/(x - ai)ⁿ

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