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f'(x) = 2xln(1+x) + x^2/(1+x) = 2xln(1+x) + x -1 + 1/(1+x), f'(0) = 0
f''(x) = 2ln(1+x) + 2x/(1+x) +1 - 1/(1+x)^2
= 2ln(1+x) + 3 - 2/(1+x) - 1/(1+x)^2, f''(0) = 0
f'''(x) = 2/(1+x) + 2/(1+x)^2 + 2!/(1+x)^3, f'''(0) = 6
f^(4)(x) = -2/(1+x)^2 - 2*2!/(1+x)^3 - 3!/(1+x)^4, f^(4)(0) = -12
f^(5)(x) = 2*2!/(1+x)^3 + 2*3!/(1+x)^4 + 4!/(1+x)^5, f^(5)(0) = 40
..........................................................................
f^(n)(x) = (-1)^(n+1)[2*(n-3)!/(1+x)^(n-2) + 2*(n-2)!/(1+x)^(n-1) + (n-1)!/(1+x)^n]
f^(n)(0) = (-1)^(n+1)[2*(n-3)! + 2*(n-2)! + (n-1)!] ( n ≥ 3 )
f^(n)(0) = 0 (n ≤ 2)
f''(x) = 2ln(1+x) + 2x/(1+x) +1 - 1/(1+x)^2
= 2ln(1+x) + 3 - 2/(1+x) - 1/(1+x)^2, f''(0) = 0
f'''(x) = 2/(1+x) + 2/(1+x)^2 + 2!/(1+x)^3, f'''(0) = 6
f^(4)(x) = -2/(1+x)^2 - 2*2!/(1+x)^3 - 3!/(1+x)^4, f^(4)(0) = -12
f^(5)(x) = 2*2!/(1+x)^3 + 2*3!/(1+x)^4 + 4!/(1+x)^5, f^(5)(0) = 40
..........................................................................
f^(n)(x) = (-1)^(n+1)[2*(n-3)!/(1+x)^(n-2) + 2*(n-2)!/(1+x)^(n-1) + (n-1)!/(1+x)^n]
f^(n)(0) = (-1)^(n+1)[2*(n-3)! + 2*(n-2)! + (n-1)!] ( n ≥ 3 )
f^(n)(0) = 0 (n ≤ 2)
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