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Let `g: R text(\ {−1}) -> R,\ \ g(x) = (sin(pi x))/(x + 1)`.
Evaluate `g prime(1)`. (2 marks)
`-pi/2`
| `u = sin(pi x)` | `v = x + 1` | |
| `u prime = pi cos(pi x)` | `v prime = 1` |
| `g prime(x)` | `= (v u prime – u v prime)/v^2` |
| `= ((x + 1) ⋅ pi cos(pi x) – sin (pi x))/(x + 1)^2` | |
| `g prime(1)` | `= (2 pi cos(pi) – sin(pi))/2^2` |
| `= (2 pi(-1) – 0)/4` | |
| `= -pi/2` |
Let `f(x)=(1 + tan x)^10.` Find `f′(x)` (2 marks)
`10 sec^2 x \ (1 + tan x)^9`
`f(x) = (1 + tan x)^10`
| `f′(x)` | `= 10 (1 + tan x)^9 xx d/(dx) (tan x)` |
| `= 10 sec^2 x \ (1 + tan x)^9` |
If `f(x)= 2 sin 3x - 3 tan x`, find `f′(0)`. (2 marks)
`3`
| `y` | `= 2 sin 3x – 3 tan x` |
| `(dy)/(dx)` | `= 6 cos 3x – 3 sec^2 x` |
`text(When)\ \ x = 0,`
| `(dy)/(dx)` | `= 6 cos 0 – 3 sec^2 0` |
| `= 6 (1) – 3/(cos^2 0)` | |
| `= 6 – 3` | |
| `= 3` |