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Calculus, 2ADV C4 2022 HSC 18

  1. Differentiate  `y=(x^(2)+1)^(4)`.   (2 marks)

  2. Hence, or otherwise, find `int x(x^(2)+1)^(3)dx`.   (1 mark)

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a.    `dy/dx=8x(x^2+1)^3`

 b.    `1/8(x^2+1)^4+C`

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a.    `y=(x^2+1)^4`

`text{Using chain rule:}`

`dy/dx` `=4 xx 2x(x^2+1)^3`
  `=8x(x^2+1)^3`

 

b.     `intx(x^2+1)^3\ dx` `=1/8 int8x(x^2+1)^3\ dx`
    `=1/8(x^2+1)^4+C`

Calculus, 2ADV C4 2011 HSC 4d

  1. Differentiate  `y=sqrt(9 - x^2)`  with respect to  `x`.   (2 marks)

  2. Hence, or otherwise, find  `int (6x)/sqrt(9 - x^2)\ dx`.   (2 marks)

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a.    `-x/sqrt(9-x^2)`

b.    `-6 sqrt(9-x^2) + C`

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IMPORTANT: Some students might find calculations easier by rewriting the equation as `y=(9-x^2)^(1/2)`.
a.     `y` `= sqrt(9-x^2)`
    `= (9-x^2)^(1/2)`

 

`dy/dx` `=1/2 xx (9-x^2)^(-1/2) xx d/dx (9-x^2)`
  `= 1/2 xx (9-x^2)^(-1/2) xx-2x`
  `=-x/sqrt(9-x^2)`

 

b.     `int (6x)/sqrt(9=x^2)\ dx` `=-6 int (-x)/sqrt(9-x^2)\ dx`
    `=-6 (sqrt(9-x^2)) + C`
    `=-6 sqrt(9-x^2) + C`