Final answer:
To obtain the n-th derivative of f(x) = (x - 1)⁻¹, apply the power rule and chain rule iteratively, leading to the general formula fⁿ⁺¹(x) = n! × (-1)ⁿ × (x - 1)⁻(n+1).
Step-by-step explanation:
To find the n-th derivative of f(x) = (x - 1)⁻¹, you need to use the formula for the derivative of a power of x, f'(x) = (-1) × (x - 1)⁻².
Subsequently, by applying the chain rule and power rule iteratively, you can find the second derivative, f''(x) = 2! × (x - 1)⁻³, and so on.
Each time you take the derivative, you multiply by the new exponent (which is negative and decrements by 1) and increase the factorial coefficient.
The general n-th derivative will be f(n)(x) = n! × (-1)n × (x - 1)⁻(n+1).