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)^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 ]
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