smc-967-30-Product Rule

Calculus, 2ADV C2 SM-Bank 2 MC

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

\(\log _e 10+\log _e x\)
\(\dfrac{\log _e 10+1}{\log _e 10}\)
\(\dfrac{1}{\log _e 10}+\log _x 10\)
\(\dfrac{1}{\log _e x}+\log _{10} x\)

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\(\Rightarrow B\)

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

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

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

\(\Rightarrow B\)

Calculus, 2ADV C3 2024 MET1 1a

Let \(y=e^x \cos\,3 x\).

Find \(\dfrac{d y}{d x}\) (1 mark)

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

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

Calculus, 2ADV C2 EQ-Bank 4 MC

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

`2/(x^(2x)ln2`
`2/ln2 + 2log_2x`
`log_2x+2/ln2`
`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 SM-Bank 8

Let `y= (x + 5) log_e (x)`.

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

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`log_e 5 +2`

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

`:. dy/dx|_(x=5)=log_e 5 +2`

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)`

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

`y`	`= 5^(x^2) * 5x`
`(dy)/(dx)`	`= ln52x5^(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 3

Differentiate `3x 6^x`. (2 marks)

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

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COMMENT: See HSC exam reference sheet when differentiating `6^x`.

`y`	`= 3x * 6^x`
`(dy)/(dx)`	`= 36^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`

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`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`

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

Calculus, 2ADV C2 2012 HSC 12ai

Differentiate with respect to `x`

`(x-1)log_ex` (2 marks)

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`log_ex+1-1/x`

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`y`	`=(x-1)log_ex`
`dy/dx`	`=1(log_ex)+(x-1)1/x`
	`=log_ex+1-1/x`

Calculus, 2ADV C2 2013 HSC 11d

Differentiate `x^2e^x` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

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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)`