For n = 1
LHS = d/dx (1/1-x)
LHS = -1.(1-x)^2.-1
LHS = 1/(1-x)^2
RHS = 1!/(1-x)^2
RHS = 1/(1-x)^2
Hence the statement is true for n = 1
Assuming the statement is true for n = k
d^k/dx^k (1/1-x) = k!/[(1-x)^k]
For n = k + 1
d^k+1/dx^k+1 (1/1-x) = (k+1)!/[(1-x)^k+1]
LHS = d/dx...