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Find `f′(x)`, where `f(x) = (x^2 + 3)/(x - 1).` (2 marks)
`((x – 3) (x + 1))/(x – 1)^2`
`f(x) = (x^2 + 3)/(x – 1)`
`text(Using the quotient rule)`
| `u` | `= x^2 + 3` | `\ \ \ \ \ \ v` | `= x – 1` |
| `u prime` | `= 2x` | `\ \ \ \ \ \ v prime` | `= 1` |
| `f′(x)` | `= (u prime v – uv prime)/v^2` |
| `= (2x (x – 1) – (x^2 + 3) xx 1)/(x – 1)^2` | |
| `= (2x^2 – 2x – x^2 – 3)/(x – 1)^2` | |
| `= (x^2 – 2x – 3)/(x – 1)^2` | |
| `= ((x – 3) (x + 1))/(x – 1)^2` |
Differentiate `(x + 2)/(3x-4).` (2 marks)
`(-10)/(3x-4)^2`
`y = (x + 2)/(3x-4)`
`text(Using the quotient rule:)`
| `(g/h)^{′}` | `= (g^{′} h-gh^{′})/h^2` |
| `y^{′}` | `= (1 (3x-4)-(x + 2) · 3)/(3x-4)^2` |
| `= (-10)/(3x-4)^2` |
Let `f: (-2, oo) -> R,\ f(x) = x/(x + 2)`.
Differentiate `f` with respect to `x`. (2 marks)
`f prime(x) = 2/(x + 2)^2`
`text(Using Quotient Rule:)`
| `(h/g)′` | `= (h′ g – h g′)/g^2` |
| `:. f prime (x)` | `= (1 xx (x + 2) – x xx 1)/(x + 2)^2` |
| `= 2/(x + 2)^2` |