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Let `f(x) = x^2 cos(3x)`.
Find `f^{′}(pi/3)`. (2 marks)
`-(2pi)/3`
| `f(x)` | `= x^2 cos 3x` | |
| `f^{′}(x)` | `= x^2 ⋅ 3(-sin 3x) + 2x cos 3x` | |
| `f^{′}(pi/3)` | `= (pi/3)^2 ⋅ 3 (-sin pi) + 2 (pi/3) cos pi` | |
| `= -(2pi)/3` |
Differentiate `x^2 sin x`. (2 marks)
`x^2 ⋅ cos x + 2x sin x`
`text(Using the product rule:)`
`d/(dx) (x^2 sin x) = x^2 ⋅ cos x + 2x sin x`
Differentiate `x tan x` with respect to `x`. (2 marks)
`dy/dx = x sec^2 x + tan x `
i. `y = x tan x`
`text(Using product rule)`
| `d/dx (uv)` | `= u prime v + uv prime` |
| `:.dy/dx` | `= tan x + x xx sec^2 x` |
| `= x sec^2 x + tan x` |
Differentiate with respect to `x`:
`x sin x` (2 marks)
| `y` | `= x sin x` |
| `dy/dx` | `= x cos x + sin x xx 1` |
| `= x cos x + sin x` |
Differentiate `x^2 tan x` with respect to `x`. (2 marks)
`2x tanx + x^2 sec^2 x`
`y = x^2 tan x`
`text(Using product rule:)`
| `d/dx (uv)` | ` = u prime v + u v prime` |
| `dy/dx` | `=2x tanx + x^2 sec^2 x` |