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Calculus, 2ADV C2 EQ-Bank 26

Show that  \(\dfrac{d}{d x}(\cot x)=-\operatorname{cosec}^2 x\).   (2 marks)

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\(\text{See Worked Solutions}\)

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\(\cot\,x=\dfrac{\cos\,x}{\sin\,x}\)

\(\text{Using the quotient rule:}\)

\(\dfrac{d}{d x}\left(\dfrac{\cos x}{\sin x}\right)\) \(=\dfrac{-\sin\, x\,\sin\, x-\cos\, x\, \cos \,x}{\sin ^2 x}\)  
  \(=\dfrac{-\sin ^2 x-\cos ^2 x}{\sin ^2 x}\)  
  \(=-\left(\dfrac{1}{\sin ^2 x}\right)\)  
  \(=-\operatorname{cosec}^2 x\)  

Calculus, 2ADV C2 EQ-Bank 19

  1. Prove that  \(\cos x+\sin x\, \tan x=\sec x\).   (1 mark)

  2. Hence, or otherwise, find  \(\dfrac{d}{dx}\left(\dfrac{1}{\cos x+\sin x \,\tan x}\right)\).   (2 marks)

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a.    \(\text{Proof (See worked solutions)}\)

b.    \(-\sin x\)

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a.    \(\text{Prove}\ \ \cos x+\sin x\, \tan x=\sec x\)

\(\text{LHS}\) \(=\cos x+\sin x \cdot \dfrac{\sin x}{\cos x}\)
  \(=\dfrac{\cos ^2 x+\sin ^2 x}{\cos x}\)
  \(=\dfrac{1}{\cos x}\)
  \(=\sec x=\text{RHS}\)

 

b.     \(\begin{array}{r} \dfrac{d}{d x}\end{array} \left(\dfrac{1}{\cos x+\sin x\, \tan x}\right)\) \(=\dfrac{d}{d x}\left(\dfrac{1}{\sec x}\right)\)
    \(=\begin{array}{r}\dfrac{d}{dx}\end{array} \left(\cos x\right)\)
    \(=-\sin x\)

Calculus, 2ADV C3 2024 MET1 1a

Let  \(y=e^x \cos\,3 x\).

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

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

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\(y\) \(=e^x \cos(3x)\)
\(\dfrac{dy}{dx}\) \(=e^x.(-3\sin(3x))+\cos(3x).e^x\)
  \(=e^x(\cos(3x)-3\sin(3x))\)

Calculus, 2ADV C2 2022 HSC 25

Let  `f(x)=sin(2x)`.

Find the value of  `x`, for  `0 < x < pi`, for which  `f^(')(x)=-sqrt3`  AND  `f^('')(x)=2`.   (3 marks)

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`(7pi)/12`

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`f^(′)(x)=2cos(2x)`

`2cos(2x)` `=-sqrt3`
`cos(2x)` `=-sqrt3/2`
`2x` `=pi-pi/6,\ \ pi+pi/6`
  `=(5pi)/6,\ \ (7pi)/6`
`x` `=(5pi)/12,\ \ (7pi)/12`

 
`f^(″)(x)=-4sin(2x)`

`-4sin(2x)` `=2`
`sin(2x)` `=-1/2`
`2x` `=pi+pi/6,\ \ 2pi-pi/6`
  `=(7pi)/6,\ \ (11pi)/6`
`x` `=(7pi)/12,\ \ (22pi)/12`

 
`:.x=(7pi)/12\ \ text{(satisfies both equations)}`


Mean mark 53%.

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 2016 HSC 5 MC

What is the derivative of  `ln (cos x)?`

  1. `-sec x`
  2. `-tan x`
  3. `sec x`
  4. `tan x`
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`B`

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`y` `= ln (cos x)`
`(dy)/(dx)` `= (-sin x)/(cos x)`
  `= -tan x`

  
`=>  B`

Calculus, 2ADV C2 2010 HSC 2a

Differentiate  `cosx/x`  with respect to  `x`.   (2 marks) 

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 `(-x sinx-cos x)/(x^2)`

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MARKER’S COMMENT: A significant number of students did not know the quotient rule and many found assistance by writing out `u,\ u prime,\ v,\ v prime\ ` before going into calculations.

`y = cos x/x`

`text(Let)` `\ \ u = cos x` `v = x`
  `\ \ u^{prime} =-sin x` `v ^{prime} = 1`

  
`text(Using quotient rule:)`

`dy/dx` `= (u^{prime} v-u v^{prime})/(v^2)`
  `= (-sinx *x-cos x*1)/x^2`
  `= (-x sin x-cos x)/(x^2)`

Calculus, 2ADV C2 2012 HSC 12aii

Differentiate with respect to  `x`. 

`(cos x)/(x^2)`.   (2 marks)

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`(-x sin x-2 cos x)/(x^3)`

 

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`y = cosx/(x^2)`

`u = cos x` `\ \ \ \ \ \ v = x^2`
`u^{prime} =-sin x` `\ \ \ \ \ \ v^{prime}= 2x`

 

`text(Using the quotient rule,)` 

`dy/dx` `= (vu^{prime}\-uv^{prime})/(v^2)`
  `= (x^2*-sin x-cos x*2x )/(x^4)`
  `= (-x sin x-2 cos x)/(x^3)`

Calculus, 2ADV C2 2013 HSC 4 MC

What is the derivative of  `x/cosx`?

  1. `(cosx+xsinx)/(cos^2 x)`  
  2. `(cosx-xsinx)/(cos^2 x)`  
  3. `(xsinx-cosx)/(cos^2 x)`
  4. `(-xsinx-cosx)/(cos^2 x)`
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`A`

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`y = x/cosx`

`text(Let)\ \ \ \ ` `u = x\ \ \ \ \ \ \ ` `v = cosx`
  `u ^ {prime} = 1\ \ \ \ \ \ \ ` `v^{prime} =-sin x`

 

`:.\ dy/dx` `= (vu ^ {prime}\-uv ^ {prime})/v^2`
  `= (cosx xx 1-x (-sinx))/(cosx)^2`
  `= (cosx + xsinx)/(cos^2x)`

 
`=>  A`