please answer questions 4 part (a, b, and c ) and please write on front each ans

Question

please answer questions 4 part (a, b, and c ) and please write on front each answer, the letter that relate to it .

4. ay using the rate.compute and the second darmvative the quanewt simpury of function by Use long divisios in reunu the function f(r) i ue som coeupule and simplify the sizhderivuive this function (Note: You may also wish by vrrify the second drri valve obtained in part (alusist dis approach c) Find the general formala for the drivasve of se function fo write your simpuned ansuer using facorial acation

Explanation / Answer

From the given question,

according to quotient rule, if y= u/v, then dy/dx= (vu'-uv')/(v2)

f(x)=x2/(1-x)

=(1-x)(2x)-(x2)(-1)/(1-x)2

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

=(2x-x2)/((1-x)2

d2y/dx2=[(1-x)2(2-2x)-(2x-x2)(2)(1-x)(-1)]/(1-x)4

=[2(1-x)3+2x(2-x)(1-x)]/(1-x)4

=2(1-x)[(1-x)2+2x(2-x)]/(1-x)4

=2[1-2x+x2+4x-2x2]/(1-x)3

=2(1+2x-x2]/(1-x)3

(b) f(x)= x6/(1-x)

1-x6/1-x= x5+x4+x3+x2+x+1

1/ (1-x) - x6/1-x=x5+x4+x3+x2+x+1

x6/1-x = 1/ (1-x) - [x5+x4+x3+x2+x+1]

y=1/ (1-x) - [x5+x4+x3+x2+x+1]

dy/dx= (1)/(x-1)2-[5x4+4x3+3x2+2x+1]

d2y/dx2=(-2)/(x-1)3

.....

d6y/dx6=(-1)(6!)/(x-1)7

(c) general formula for nth derivative

dny/dxn=(-1)(n!)/(x-1)n+1

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