# Differentiation rule for power functions

From Calculus

## Statement

We have the following differentiation rule:

where is a constant. Some notes on the validity:

Case on | Values of real for which this makes sense |
---|---|

all nonzero . Also makes sense at if we interpret the left side as 1 (constant equal to the list at 0) and the right side as 0. | |

a rational number with odd denominator and greater than or equal to 1 | All |

a real number greater than 1 that is not rational with odd denominator | All . One-sided derivative makes sense at 0. |

a rational number with odd denominator and between 0 and 1 | All . At 0, we have a vertical tangent or vertical cusp depending on the numerator of the rational function. |

a real number between 0 and 1 that is not rational with odd denominator | All . One-sided vertical tangent at 0. |

a rational number with odd denominator and less than 0 | All . At 0, we have a vertical asymptote |

a real number less than 0 that is not rational with odd denominator | All . One-sided vertical asymptote at 0. |