Question

find the nth derivative of the function 6/(x^2-4) (x-1) ?
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Answer 1

Answer:

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Step-by-step explanation:

y=x4(x−1)(x−2)=Ax2+Bx+C+Dx−1+Ex−2  

⟹x4=Ax2(x−1)(x−2)+Bx(x−1)(x−2)+C(x−2)(x−1)+D(x−2)+E(x−1)  

⟹x4=Ax4+(B−3A)x3+x2(2A−3B+C)+(2B−3C+D+E)x+(2C−2D−E)  

x4:A=1  

x3:B−3A=0⟹B=3A=3  

x2:2A−3B+C=0⟹C=3B−2A=9−2=7  

x:2B−3C+D+E=0⟹E=−2B+3C−D=15−D  

Constant: 2C−2D−E=0⟹14−2D−15+D=0  

⟹D=−1⟹E=15−D=16  

∴y=x4(x−1)(x−2)=x2+3x+7−1x−1+16x−2  

∴y=x4(x−1)(x−2)=x2+3x+7−1x−1+16x−2  

⟹y=x2+3x+7−(x−1)−1+16(x−2)−1  

y′=2x+3+(x−1)−2−16(x−2)−2  

y′′=2+(−2)(x−1)−3−16(−2)(x−2)−3  

=2+(−1)2⋅2!⋅[16(x−2)−3−(x−1)−3]  

y′′′=(−1)3⋅3!⋅[16(x−2)−4−(x−1)−4]  

y′n=(−1)n⋅n!⋅[16(x−2)−n−1−(x−1)−n−1] for n > 3

Answer 2

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