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Evaluate `f^{′}(1)`, where `f(x) = x^2 / sqrt(2x + 3)`. (4 marks)
`9 / (5sqrt5)`
`f(x) = x^2(2x + 3)^(-1/2)`
`f^{′}(x)` `= 2x(2x + 3)^(-1/2) + x^2(-1/2)(2x + 3)^(-3/2)(2)`
`= (2x)/(sqrt(2x + 3)) – (x^2)/(2x + 3)^(3/2)`
`= [2x(2x + 3) – x^2] / (2x + 3)^(3/2)`
`= (3x^2 + 6x) / (2x + 3)^(3/2)`
`f^{′}(1)` `= (3(1)^2 + 6(1)) / (2(1) + 3)^(3/2)`
`= 9 / (5sqrt5)`
Differentiate `2x(1-4x)^5` with respect to `x`. (2 marks) --- 5 WORK AREA LINES (style=lined) --- `y^{′}=2(1-4x)^4(1-24x)` `y=2x(1-4x)^5` `text{Using the product and chain rules:}`
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`y^{′}`
`=2 xx (1-4x)^5-40x(1-4x)^4`
`=2(1-4x)^4(1-4x-20x)`
`=2(1-4x)^4(1-24x)`
Let `g(x) = (2-x^3)^3`.
Evaluate `g^{′}(-1)`. (2 marks)
`-81`
`text(Using Chain Rule:)`
| `g^{′}(x)` | `= 3 (2-x^3)^2 (-3x^2)` |
| `= -9x^2 (2-x^3)^2` | |
| `:. g^{′}(1)` | `= -9 (–1)^2 [2-(–1)^3]^2` |
| `= -81` |