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The Power Rule - Calculus - 04 Jun 2018 at 02:06

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The Power Rule - Calculus

The Power Rule in Calculus is used when finding derivatives, and can be simplified to the following steps:

  1. Multiply the term by the exponent (power)
  2. Subtract 1 from the exponent

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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