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Calculus, 2ADV C2 EQ-Bank 27

Differentiate  `pi^(2x)`.   (2 marks)

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`2log_e(pi) * pi^(2x)`

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COMMENT: `pi` is a constant. i.e. differentiate `a^(f(x))`.
`y` `=pi^(2x)`
`dy/dx` `=log_e(pi) * 2 * pi^(2x)`
  `=2log_e(pi) *pi^(2x)`

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`

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`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 28

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

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

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`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 25

Differentiate  `3x  6^x`.   (2 marks)

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`3*6^x(xln6 +1)`

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`y= 3x * 6^x`

`text{Using the product rule:}`

`(dy)/(dx)= 3*6^x + ln6 * 6^x *3x= 3*6^x(1 + xln6)`

Calculus, 2ADV C2 EQ-Bank 26

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)`

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`y = 10^(5x^2-3x)`

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

Calculus, 2ADV C2 EQ-Bank 15

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

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`2/(xln2)`

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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)`