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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))`
Show Answers Only

`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 SM-Bank 4

Differentiate  `5^(x^2)5x`.  (2 marks)

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`5^(x^2 + 1)(ln5*2x^2 + 1)`

Show Worked Solution

COMMENT: See HSC exam reference sheet when differentiating  `5^x`.

`y` `= 5^(x^2) * 5x`
`(dy)/(dx)` `= ln5*2x*5^(x^2)*5x + 5^(x^2)*5`
  `=5^(x^2)(ln5*10x^2 + 5)`
  `=5^(x^2 + 1)(ln5*2x^2 + 1)`

Calculus, 2ADV C2 EQ-Bank 2

Differentiate with respect to `x`:

`10^(5x^2 - 3x)`.  (2 marks)

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`(dy)/(dx) = ln 10  (10x – 3) * 10^(5x^2 – 3x)`

Show Worked Solution

`y = 10^(5x^2 – 3x)`

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

`(dy)/(dx) = ln 10  (10x – 3) * 10^(5x^2 – 3x)`