If f ⁡ x = 1 + x a 1 b 1 1 + x a 1 b 2 1 + x a 1 b 3 1 + x a 2 b 1 1 + x a 2 b 2 1 + x a 2 b 3 1 + x a 3 b 1 1 + x a 3 b 2 1 + x a 3 b 3 , find the coefficient of x in the expansion of f(x)

Step-by-step solution

[Remember if f(x) = A 0 + A 1 x + A 2 x 2 + A 3 x 3 +... then f'(x) = A 1 + 2A 2 x + 3A 3 x 2 + ... f''(x) = 2A 2 + 6A 3 x + ... ∴ f'(0) = A 1 = coefficient of x. f''(0) = 2A 2 = 2 × (coefficient of x 2 ), etc.] Here f'(x) = a 1 b 1 1 + x a 1 b 1 - 1 a 1 b 2 1 + x a 1 b 2 - 1 a 1 b 3 1 + x a 1 b 3 - 1 1 + x a 2 b 1 1 + x a 2 b 2 1 + x a 2 b 3 1 + x a 3 b 1 1 + x a 3 b 2 1 + x a 3 b 3 + 1 + x a 1 b 1 1 + x a 1 b 2 1 + x a 1 b 3 a 2 b 1 1 + x a 2 b 1 - 1 a 2 b 2 1 + x a 2 b 2