## Question

###### Suppose that k is a positive integer constant. Let f(x) = 1/(x^k-1). The nth derivative of this function is of the form f^(n) (x) =Pn(x)/(x^k -1)n+1 where Pn(x)is some polynomial function. it is easy to see thatP0(x) = 1 since we have differentiated zerotimes in thiscase. Also P1(x) = -kxk-1 a) suppose n>= 1. Find a formula for Pn(x)which depends only on x, n, k, Pn-1(x), andPn-1'(x).b) Find the value of Pâ‚â‚€â‚€â‚(1). Explain how youarrived at your answer.

Suppose that k is a positive integer constant. Let f(x) = 1/(x^k -1). The nth derivative of this function is of the form f^(n) (x) = Pn(x)/(x^k -1)n+1 where Pn(x) is some polynomial function. it is easy to see that P0(x) = 1 since we have differentiated zerotimes in this case. Also P1(x) = -kxk-1 a) suppose n>= 1. Find a formula for Pn(x) which depends only on x, n, k, Pn-1(x), and Pn-1'(x). b) Find the value of Pâ‚â‚€â‚€â‚(1). Explain how you arrived at your answer.

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