# Power Rule

The Power Rule, one of the most commonly used rules in Calculus, says:

The derivative of x^{n} is nx^{(n-1)}

### Example: What is the derivative of x^{2} ?

For x^{2} we use the Power Rule with **n=2**:

The derivative of x^{2} | = | 2x^{(2-1)} |

= | 2x^{1} | |

= | 2x |

Answer: the derivative of **x ^{2}** is

**2x**

"The derivative of" can be shown with the little mark ’

So we get this definition:

f’(x^{n}) = nx^{(n-1)}

### Example: What is the derivative of x^{3} ?

f’(x^{3}) = 3x^{3−1} = **3x ^{2}**

"The derivative of" can also be written \frac{d}{dx}

### Example: What is (1/x) ?

1/x is also **x ^{-1}**

Using the Power Rule with **n = −1**:

x^{n} = nx^{n−1}

x^{−1} = −1x^{−1−1} = **−x ^{−2}**

## How to Remember

"multiply by power

then reduce power by 1"

## A Short Table

Here is the Power Rule with some sample values. See the pattern?

f | f’(x^{n}) = nx^{(n-1)} | f’ |
---|---|---|

x | 1x^{(1-1)} = x^{0} | 1 |

x^{2} | 2x^{(2-1)} = 2x^{1} | 2x |

x^{3} | 3x^{(3-1)} = 3x^{2} | 3x^{2} |

x^{4} | 4x^{(4-1)} = 4x^{3} | 4x^{3} |

etc... | ||

And for negative exponents: | ||

x^{-1} | -1x^{(-1-1)} = -x^{-2} | -x^{-2} |

x^{-2} | -2x^{(-2-1)} = -2x^{-3} | -2x^{-3} |

x^{-3} | -3x^{(-3-1)} = -3x^{-4} | -3x^{-4} |

etc... |