SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV C2 SM-Bank 1 MC

If  `f(x)=e^(g(x^(2)))`, where `g` is a differentiable function, then  `f^(′)(x)`  is equal to

  1. `2xe^(g(x^(2)))`
  2. `2xg(x^(2))e^(g(x^(2)))`
  3. `2xg^(′)(x^(2))e^(g(x^(2))`
  4. `2xg^(′)(2x)e^(g(x^(2)))`
Show Answers Only

`C`

Show Worked Solution

`f(x)=e^(g(x^2))`

`text{Using the chain rule (twice):}`

`f^(′)(x)` `=d/dx[g(x^2)] * e^(g(x^2))`  
  `=2x*g^(′)(x^2)*e^(g(x^2))`  

 
`=> C`

Calculus, 2ADV C2 EQ-Bank 2

Differentiate with respect to  `x`: 

`e^(tan(2x))`   (2 marks)

Show Answers Only

 `2 sec^2(2x)* e^(tan(2x))`

Show Worked Solution
`y` `=e^(tan(2x))`
`dy/dx` `= d/(dx)tan(2x) xx e^(tan(2x))`
  `= 2 sec^2(2x)* e^(tan(2x))`

Calculus, 2ADV C2 2019 MET1 1a

Let  `y = (2e^(2x) - 1)/e^x`.

Find  `(dy)/(dx)`.  (2 marks)

Show Answers Only

`(dy)/(dx) = 2e^x + e^(-x)`

Show Worked Solution

`text(Method 1)`

`y` `= 2e^x – e^(-x)`
`(dy)/(dx)` `= 2e^x + e^(-x)`

 

`text(Method 2)`

`(dy)/(dx)` `= (4e^(2x) ⋅ e^x – (2e^(2x) – 1) e^x)/(e^x)^2`
  `= (4e^(3x) – 2e^(3x) + e^x)/e^(2x) `
  `= (2e^(2x) + 1)/e^x`

Calculus, 2ADV C2 SM-Bank 4

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

Show Answers Only

`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)

Show Answers Only

`(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)`

 

Calculus, 2ADV C4 2018 HSC 5 MC

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

  1. `cos (1/x)`
  2. `cos (ln x)`
  3. `cos ((ln x)/x)`
  4. `(cos (ln x))/x`
Show Answers Only

`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 2015 HSC 11e

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

Show Answers Only

`5 (e^x + 1) (e^x + x)^4`

Show Worked Solution
`y` `= (e^x + x)^5`
 `(dy)/(dx)` `= 5 (e^x + x)^4 xx d/(dx) (e^x + x)`
  `= 5 (e^x + x)^4 xx (e^x + 1)`
  `= 5 (e^x + 1) (e^x + x)^4`

Calculus, 2ADV C2 2012 HSC 11d

Differentiate    `(3+e^(2x))^5`.    (2 marks) 

Show Answer Only

`10e^(2x)(3+e^(2x))^4`

Show Worked Solutions

`y=(3+e^(2x))^5`

`(dy)/dx` `=5(3+e^(2x))^4 xx  d/(dx)(3+e^(2x))`
  `=5(3+e^(2x))^4 xx 2e^(2x)`
  `=10e^(2x)(3+e^(2x))^4`