Find f'(x) and f"(x): f(x)=x/x^2-1

Question

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Answer 1

To differentiate the ratio of two functions, follow the Quotient Rule: f'(x) = [(bottom function times derivative of top function) minus (top function times derivative of bottom function)] divided by [square of bottom function].

For f(x) = x/(x² − 1), f'(x) = [((x² − 1) × (1) − (x) × (2x))] ÷ [(x² − 1)²] or [x² − 1) − 2x²]/[(x² − 1)²] which simplifies to -(x² + 1)/(x² − 1)².

Then f"(x) is the first derivative of f'(x) equal to -(x² + 1)/(x² − 1)²: [(x² − 1)² × (-2x) − -(x² + 1) × 2(2x)(x² − 1)] ÷

[(x² − 1)⁴] or [(-2x)(x² − 1) + (x² + 1)(4x)]/ [(x² − 1)³] which simplifies to (2x³ + 6x)/(x² − 1)³.