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Calculus, 2ADV C2 2019 MET1 1b

Let  `f(x) = x^2 cos(3x)`.
 
Find  `f^{prime}(pi/3)`.   (2 marks)

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`-(2pi)/3`

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`f(x)` `= x^2 cos 3x`
`f^{prime}(x)` `= x^2 ⋅ 3(-sin 3x) + 2x cos 3x`
`f^{prime}(pi/3)`  `= (pi/3)^2 ⋅ 3 (-sin pi) + 2 (pi/3) cos pi`
  `= -(2pi)/3`

Calculus, 2ADV C2 2006 HSC 2ai

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

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`dy/dx = x  sec^2 x + tan x `

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`y = x tan x`

`text(Using product rule)`

`d/dx (uv)` `=uv ^{prime}+ u^{prime}v`
`:.dy/dx` `=x xx sec^2 x+ 1 xxtan x `
  `= x sec^2 x + tan x`

Calculus, 2ADV C2 2010 HSC 1e

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

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

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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`