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Let `g: R text(\ {−1}) -> R,\ \ g(x) = (sin(pi x))/(x + 1)`.
Evaluate `g^{prime}(1)`. (2 marks)
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`-pi/2`
| `u = sin(pi x)` | `v = x + 1` | |
| `u^{prime}=pi cos(pi x)` | `v^{prime}=1` |
| `g^{prime}(x)` | `=(vu^{prime}-uv^{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^{\prime}(x)`. (2 marks)
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`10 sec^2 x \ (1 + tan x)^9`
`f(x) = (1 + tan x)^10`
| `f^{\prime}(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^{prime}(0)`. (2 marks)
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`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` |