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Calculus, 2ADV C2 SM-Bank 2 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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\(\Rightarrow B\)

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 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)`