smc-967-20-Logs

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 6

Differentiate with respect to `x`:

`log_e x^x`. (2 marks)

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`1 + log_ex`

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

Calculus, 2ADV C2 EQ-Bank 1

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)`

Calculus, 2ADV C4 2018 HSC 5 MC

What is the derivative of `sin(ln x),` where `x > 0`?

`cos (1/x)`
`cos (ln x)`
`cos ((ln x)/x)`
`(cos (ln x))/x`

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`D`

Show Worked Solution

`y`	`= sin (ln x)`
`(dy)/(dx)`	`= cos (ln x) xx d/(dx) (ln x)`
	`= cos (ln x) xx 1/x`
	`= (cos (ln x))/x`

`=> D`

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 2016 HSC 5 MC

What is the derivative of `ln (cos x)?`

`– sec x`
`– tan x`
`sec x`
`tan x`

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`B`

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`y`	`= ln (cos x)`
`(dy)/(dx)`	`= (-sin x)/(cos x)`
	`= -tan x`

`=> B`

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 C3 2005 HSC 2d

Find the equation of the tangent to `y = log_ex` at the point `(e, 1)`. (2 marks)

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

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`y = x/e`

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`y = log_ex`

`dy/dx = 1/x`

`text(At)\ \ (e, 1),`

`m = 1/e`

`text(Equation of tangent,)\ \ m = 1/e,\ text(through)\ (e, 1)`

`y – y_1`	`= m(x – x_1)`
`y – 1`	`= 1/e(x -e)`
`y – 1`	`= x/e – 1`
`y`	`= x/e`

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 2011 HSC 1d

Differentiate `ln(5x+2)` with respect to `x`. (2 marks)

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`5/(5x+2)`

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`y`	`=ln(5x+2)`
`dy/dx`	`=5/(5x+2)`

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`