Question

Find the nth derivative of 1/(x²-6x+8)​

Answer 1

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)^n![ 1/(x+2)^n+1 -1/(x+4)^n+1 ]

Final answer:

The nth derivative is y_(n) = 2(-1)^n![1/(x+2)^n+1 -1/(x+4)^n+1 ]