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Differentiate `pi^(2x)`. (2 marks)
`2log_e(pi) * pi^(2x)`
| `y` | `=pi^(2x)` | |
| `dy/dx` | `=log_e(pi) * 2 * pi^(2x)` | |
| `=2log_e(pi) *pi^(2x)` |
Differentiate `5^(x^2)5x`. (2 marks)
`5^(x^2 + 1)(ln5*2x^2 + 1)`
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)` |
Differentiate `3x 6^x`. (2 marks)
`3*6^x(xln6 +1)`
COMMENT: See HSC exam reference sheet when differentiating `6^x`.
| `y` | `= 3x * 6^x` |
| `(dy)/(dx)` | `= 3*6^x + ln6 * 6^x *3x` |
| `= 3*6^x(1 + xln6)` |
Differentiate with respect to `x`:
`10^(5x^2 - 3x)`. (2 marks)
`(dy)/(dx) = ln 10 (10x – 3) * 10^(5x^2 – 3x)`
`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)`