smc-7129-40-Product Rule

Calculus, 2ADV C3 2024 MET1 1a

Let \(y=e^x \cos\,3 x\).

Find \(\dfrac{d y}{d x}\) (2 marks)

Show Answers Only

\(e^x (\cos(3x)-3\sin(3x))\)

Show Worked Solution

Let `f(x) = x^2 cos(3x)`.

Find `f^{prime}(pi/3)`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`-(2pi)/3`

Show Worked Solution

Differentiate `x^2 sin x`. (2 marks)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`x^2 ⋅ cos x + 2x sin x`

Show Worked Solution

`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)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`dy/dx = x sec^2 x + tan x `

Show Worked Solution

`y = x tan x`

`text(Using product rule)`

Differentiate with respect to `x`:

`x sin x`. (2 marks)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

Differentiate `x^2 tan x` with respect to `x`. (2 marks)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`2x tanx + x^2 sec^2 x`

Show Worked Solution

COMMENT: There is no necessity to take out the common factor `x` in this case.

`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`