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Find the nth derivative of 1/(x2+6x+8) |
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Answer» Given: y = 1 / (x²+6x+8) To find: The NTH derivative Calculation: => y = 1 / (x²+6x+8) y = 1 / (x+2)(x+4) y = 2[ 1/(x+2) - 1/(x+4) ] DERIVATE both SIDES of the equation w.r.t x => y₁ = dy/dx = 2[ -1/(x+2)² + 1/(x+4)² ] = -2[ 1/(x+2)² - 1/(x+4)² ] Derivate both sides of the equation w.r.t x => y₂ = d²y/dx² = -2[ -2/(x+2)³ + 2/(x+4)³ ] = 4[ 1/(x+2)³ - 1/(x+4)³ ] Derivate both sides of the equation w.r.t x =>y₃ = d³y/dx³ = 4[ -3/(x+2)⁴ + 3/(x+4)⁴ ] = -12[ 1/(x+2)⁴ - 1/(x+4)⁴ ] y_(n) = 2(-1)ⁿn![ 1/(x+2)ⁿ⁺¹ - 1/(x+4)ⁿ⁺¹ ] Final answer: The nth derivative is y_(n) = 2(-1)ⁿn![ 1/(x+2)ⁿ⁺¹ - 1/(x+4)ⁿ⁺¹ ] |
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