Questions on differentiation (Division/Quotient Rule)

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1. \(f(x) = \frac{x}{x+1}\)\(f'(x) = \frac{1}{(x+1)^2}\)
2. \(f(x) = \frac{2x}{x^2 + 1}\)\(f'(x) = \frac{2(1 – x^2)}{(x^2 + 1)^2}\)
3. \(f(x) = \frac{x^2}{x + 2}\)\(f'(x) = \frac{x^2 + 4x}{(x + 2)^2}\)
4. \(f(x) = \frac{3x + 1}{x – 1}\)\(f'(x) = \frac{-4}{(x – 1)^2}\)
5. \(f(x) = \frac{x^3}{x^2 + 1}\)\(f'(x) = \frac{x^4 + 3x^2}{(x^2 + 1)^2}\)
6. \(f(x) = \frac{x + 1}{x^2 + 1}\)\(f'(x) = \frac{-x^2 – 2x + 1}{(x^2 + 1)^2}\)
7. \(f(x) = \frac{2x^2}{x^3 + 1}\)\(f'(x) = \frac{2x(x^3 + 1) – 6x^4}{(x^3 + 1)^2}\)
8. \(f(x) = \frac{x^2 + 1}{x^2 – 1}\)\(f'(x) = \frac{-4x}{(x^2 – 1)^2}\)
9. \(f(x) = \frac{x}{x^2 + 2x + 1}\)\(f'(x) = \frac{1 – x^2}{(x^2 + 2x + 1)^2}\)
10. \(f(x) = \frac{x^3}{x^2 – 4}\)\(f'(x) = \frac{x^4 – 12x^2}{(x^2 – 4)^2}\)
11. \(f(x) = \frac{x^2 + 2x}{x^2 – 2x}\)\(f'(x) = \frac{-4x^2}{(x^2 – 2x)^2}\)
12. \(f(x) = \frac{x^2 – 1}{x^2 + 1}\)\(f'(x) = \frac{4x}{(x^2 + 1)^2}\)
13. \(f(x) = \frac{x^3 + x}{x^2 – x}\)\(f'(x) = \frac{x^4 – 2x^3 – x^2}{(x^2 – x)^2}\)
14. \(f(x) = \frac{x^2}{x^3 – 1}\)\(f'(x) = \frac{-x^4 + 2x}{(x^3 – 1)^2}\)
15. \(f(x) = \frac{x^2 + 3x}{x^2 – 3x}\)\(f'(x) = \frac{-6x^2}{(x^2 – 3x)^2}\)
16. \(f(x) = \frac{x^2 + 1}{x^3 + 1}\)\(f'(x) = \frac{-x^4 – 2x^3 + 2x}{(x^3 + 1)^2}\)
17. \(f(x) = \frac{x^3}{x^2 + 2x + 1}\)\(f'(x) = \frac{x^4 + 4x^3 + 3x^2}{(x^2 + 2x + 1)^2}\)
18. \(f(x) = \frac{x^2 – 2x}{x^2 + 2x}\)\(f'(x) = \frac{4x^2}{(x^2 + 2x)^2}\)
19. \(f(x) = \frac{x^2 + x}{x^2 – x}\)\(f'(x) = \frac{-2x^2}{(x^2 – x)^2}\)
20. \(f(x) = \frac{x^3}{x^2 + 3x + 2}\)\(f'(x) = \frac{x^4 + 6x^3 + 6x^2}{(x^2 + 3x + 2)^2}\)
21. \(f(x) = \frac{x^2 – 1}{x^3 + 1}\)\(f'(x) = \frac{-x^4 + 2x^3 + 2x}{(x^3 + 1)^2}\)
22. \(f(x) = \frac{x^2 + 2}{x^2 – 2}\)\(f'(x) = \frac{-8x}{(x^2 – 2)^2}\)
23. \(f(x) = \frac{x^3 + 2x}{x^2 – 2x}\)\(f'(x) = \frac{x^4 – 4x^3 – 4x^2}{(x^2 – 2x)^2}\)
24. \(f(x) = \frac{x^2 + 3}{x^2 – 3}\)\(f'(x) = \frac{-12x}{(x^2 – 3)^2}\)
25. \(f(x) = \frac{x^3}{x^2 + 4x + 4}\)\(f'(x) = \frac{x^4 + 8x^3 + 12x^2}{(x^2 + 4x + 4)^2}\)
26. \(f(x) = \frac{x^2 + x}{x^2 – x}\)\(f'(x) = \frac{-2x^2}{(x^2 – x)^2}\)
27. \(f(x) = \frac{x^3 + 1}{x^2 – 1}\)\(f'(x) = \frac{x^4 – 3x^2 – 2x}{(x^2 – 1)^2}\)
28. \(f(x) = \frac{x^2 – 4}{x^2 + 4}\)\(f'(x) = \frac{16x}{(x^2 + 4)^2}\)
29. \(f(x) = \frac{x^3 + 2}{x^2 – 2}\)\(f'(x) = \frac{x^4 – 6x^2 – 4x}{(x^2 – 2)^2}\)
30. \(f(x) = \frac{x^2 + 2x}{x^2 – 2x}\)\(f'(x) = \frac{-4x^2}{(x^2 – 2x)^2}\)
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