It is given that y = (1 + x)² ln(1 + x). Find d³y/dx³. Prove by mathematical induction that, for every integer n ≥ 3, dⁿy/dxⁿ = (-1)^{n-1} 2(n-3)! / (1 + x)^{n-2}.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

1. [B1] (a) Differentiates once, twice. y' = 2(1+x)ln(1+x) + (1+x)
2. [B1] y'' = 2ln(1+x) + 3
3. [B1] and three times. y''' = 2/(1+x)
4. [B1] (Allow B1 if constant term in previous line incorrect.)
5. [B1] (b) Proves base case. d³y/dx³ = (-1)² 2(0)! / (1+x) = 2/(1+x) ⇒ H3 is true.
6. [B1] States inductive hypothesis. Hk: d^k y/dx^k = (-1)^(k-1) 2(k-3)! / (1+x)^(k-2) for some k.
7. [M1] Differentiates. d^(k+1)y/dx^(k+1) = (-1)^(k-1) 2(k-3)! (-1) (k-2) (1+x)^(-(k-1))
8. [A1] = (-1)^k 2(k-2)! / (1+x)^(k-1) ⇒ Hk+1 is true
9. [A1] Proves inductive step and states conclusion. Hence by PMI Hn is true for all integers ≥ 3