04 Jun 2018
The Power Rule in Calculus is used when finding derivatives, and can be simplified to the following steps:
The result of taking the derivative of `f(x)` is often written as `f'(x)` (`f` prime of `x`).
To generalise:
`d/dx x^n = nx^(n-1)`
Some examples:
| `f` | `d/dx` | `f'` | Notes |
| `f(x^3)` |
`x^3` `= (3)x^(3-1)` `= 3x^2` |
`3x^2` | |
| `f(x^2)` |
`x^2` `= (2)x^(2-1)` `= 2x^1` `= 2x` |
`2x` |
|
| `f(x)` |
`x` `= x^1` `= (1)x^(1-1)` `= x^0` `= 1` |
`1` |
`x` is the same as `x^1` Any term raised to the power of `0` is always `1` |
| `f(x^0)` |
`x^0` `= (0)x^(0-1)` `= 0^-1` `= 0` |
`0` | |
| `f(x^-1)` |
`x^-1` `= (-1)x^(-1-1)` `= -x^-2` |
`-x^-2` or: `-1/x^2` |
This can be written either way |
| `f(x^-2)` |
`x^-2` `= (-2)x^(-2-1)` `= -2x^-3` |
`-2x^-3` or: `-2/x^3` |
This can be written either way |
| `f(x^-3)` |
`x^-3` `= (-3)x^(-3-1)` or: `= -3x^-4` |
`-3x^-4` or: `-3/x^4` |
This can be written either way |
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