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Calculus, 2ADV C2 EQ-Bank 4 MC

If  `f(x)=log_2(x^(2x))`, which expression is equal to `f^(′)(x)`?

  1. `2/(x^(2x)ln2`
  2. `2/ln2 + 2log_2x`
  3. `log_2x+2/ln2`
  4. `2/ln2 xx log_2(x^(2x-1))`
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`B`

Show Worked Solution
`f(x)` `=log_2(x^(2x))`
  `=2x log_2x`
  `=(2x lnx)/ln2`

 

`f^(′)(x)` `=1/ln2 (2x*1/x + 2lnx)`
  `=2/ln2 + (2lnx)/ln2`
  `=2/ln2 + 2log_2x`

 
`=>  B`

Calculus, 2ADV C2 EQ-Bank 13

Let  `y= (x + 5) log_e (x)`.

Find  `(dy)/(dx)`  when  `x = 5`.   (2 marks)

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`log_e 5 +2`

Show Worked Solution
`(dy)/(dx)` `= 1 xx log_e x + (x + 5) * (1)/(x)`
  `= log_e x + (x + 5)/(x)`

 
`:. \ text{when}\ x=5,\ \ dy/dx=log_e 5 +2`

Calculus, 2ADV C2 EQ-Bank 15

Differentiate  `log_2 x^2`  with respect to `x`.   (2 marks)

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`2/(xln2)`

Show Worked Solution
TIP: The new Advanced reference sheet can be used here!

`y` `= log_2 x^2`
`(dy)/(dx)` `= {:d/(dx):} ((lnx^2)/(ln2))`
  `= 1/(ln2) · d/(dx)(ln x^2)`
  `= 1/(ln2) · (2x)/(x^2)`
  `= 2/(xln2)`

Calculus, 2ADV C4 2019 HSC 13c

  1.  Differentiate `(ln x)^2`.   (2 marks)

  2.  Hence, or otherwise, find `int(ln x)/x\ dx`.   (1 mark)

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i.    `(2 ln x)/x`

ii.   `1/2 (ln x)^2 + C`

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i.    `y= (ln x)^2`

`(dy)/(dx)= 2 xx 1/x xx ln x= (2 ln x)/x`

♦ Mean mark (ii) 49%.

 

ii.   `int (ln x)/x\ dx=1/2 int (2 ln x)/x dx= 1/2 (ln x)^2 +C`

Calculus, 2ADV C2 2018 HSC 5 MC

What is the derivative of  `sin(ln x),` where  `x > 0`?

  1. `cos (1/x)`
  2. `cos (ln x)`
  3. `cos ((ln x)/x)`
  4. `(cos (ln x))/x`
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`D`

Show Worked Solution
`y` `= sin (ln x)`
`(dy)/(dx)` `= cos (ln x) xx d/(dx) (ln x)`
  `= cos (ln x) xx 1/x`
  `= (cos (ln x))/x`

  
 `=>  D`

Calculus, 2ADV C2 2017 HSC 11d

Differentiate  `x^3 ln x`.   (2 marks)

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`x^2 (3 ln\ x + 1)`

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`y = x^3 ln\ x`

`text(Using the product rule:)`

`(dy)/(dx)` `= 3x^2 * ln\ x + x^3 * 1/x`
  `= x^2 (3 ln\ x + 1)`

Calculus, 2ADV C2 2015 HSC 11f

Differentiate  `y = (x + 4) ln\ x`.   (2 marks)

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`ln\x + 4/x +1`

Show Worked Solution

`y = (x + 4) ln\ x`

`text(Using the product rule)`

`(dy)/(dx)` `= d/(dx) (x + 4) * ln x + (x + 4) d/(dx) ln\ x`
  `= ln x + (x + 4) 1/x`
  `= ln x + 4/x + 1`

Calculus, 2ADV C2 2008 HSC 2aii

Differentiate with respect to  `x`:

`x^2 log_e x`   (2 marks)

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`x + 2x log_e x`

Show Worked Solution
`y` `= x^2 log_e x`
`dy/dx` `= x^2 * 1/x + 2x * log_e x`
  `= x + 2x log_e x`

Calculus, 2ADV C4 2008 HSC 3b

  1. Differentiate  `log_e (cos x)`  with respect to  `x`.   (2 marks)

  2. Hence, or otherwise, evaluate  `int_0^(pi/4) tan x\ dx`.   (2 marks)

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a.    `-tan x`

b.    `-log_e (1/sqrt2)\ \ text(or)\ \ 0.35\ \ text{(2 d.p.)}`

Show Worked Solution
a.     `y` `= log_e (cos x)`
  `dy/dx` `= (-sin x)/(cos x)`
    `=-tan x`

 

b.     `int_0^(pi/4) tan x\ dx`
  `=-[log_e (cos x)]_0^(pi/4)`
  `=-[log_e(cos (pi/4))-log_e (cos 0)]`
  `=-[log_e (1/sqrt2)-log_e 1]`
  `=-[log_e (1/sqrt2)-0]`
  `=-log_e (1/sqrt2)`
  `= 0.346…\ = 0.35\ \ text{(2 d.p.)}`