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Calculus, 2ADV C2 EQ-Bank 7 MC

Given the function  \(f(x)=\log _{10} x^x\), which of the following expressions is equal to \(f^{\prime}(x)\) ?

  1. \(\log _e 10+\log _e x\)
  2. \(\dfrac{\log _e 10+1}{\log _e 10}\)
  3. \(\dfrac{1}{\log _e 10}+\log _x 10\)
  4. \(\dfrac{1}{\log _e x}+\log _{10} x\)
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\(B\)

Show Worked Solution

\(f(x)=\log _{10} x^x=x \log _{10} x\)

\(\text{Using product rule:}\)

\(f^{\prime}(x)\) \(=x \cdot \dfrac{1}{x \cdot \ln 10}+1 \cdot \log _{10} x\)
  \(=\dfrac{1}{\ln 10}+\log _{10} x\)
  \(=\dfrac{1}{\ln 10}+\dfrac{\ln x}{\ln 10}\)
  \(=\dfrac{\ln x+1}{\ln 10}\)

 
\(\Rightarrow B\)

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

Show Worked Solution

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

Calculus, 2ADV C2 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 28

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

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

Show Worked Solution
`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)`

Show Worked Solution

`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 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 C2 2013 HSC 11d

Differentiate  `x^2e^x`    (2 marks)

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 `xe^x(x+2)`

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`text{Using the product rule}`

`text(Let)\ \ u=x^2,\ \ \ \ \ \ u^{\ prime}=2x`

`text(Let)\ \ v=e^x,\ \ \ \ \ \ v^{\ prime}=e^x`
  

`{d(uv)}/dx` `=u^{\ prime} v+v^{\ prime} u`
  `=2x e^x +x^2 e^x `
  `=xe^x(x+2)`