### The derivative of a power function

#### Video Introduction

*The derivative of a power function.*

#### Summary

For a power function $$f(x)=x^p,$$ with exponent $p \ne 0$, its derivative is \begin{align} f'(x) = \diff{f}{x} = p x^{p-1}. \end{align} (For fractional $p$, we may need to restrict the domain to positive numbers, $x > 0$, so that the function is real valued.)

Using this formula, we calculate derivatives for small positive and negative powers as well as some fractional powers.

$p$ | $f(x)$ | $f'(x)$ or $\diff{f}{x}$ |
---|---|---|

$1$ | $x$ | $1$ |

$2$ | $x^2$ | $2x$ |

$3$ | $x^3$ | $3x^2$ |

$4$ | $x^4$ | $4x^3$ |

$p$ | $f(x)$ | $f'(x)$ or $\diff{f}{x}$ |
---|---|---|

$-1$ | $x^{-1}=\frac{1}{x}$ | $-x^{-2}=-\frac{1}{x^2}$ |

$-2$ | $x^{-2}=\frac{1}{x^2}$ | $-2x^{-3}=-\frac{2}{x^3}$ |

$-3$ | $x^{-3}=\frac{1}{x^3}$ | $-3x^{-4}=-\frac{3}{x^4}$ |

$-4$ | $x^{-4}=\frac{1}{x^4}$ | $-4x^{-5}=-\frac{4}{x^5}$ |

$p$ | $f(x)$ | $f'(x)$ or $\diff{f}{x}$ |
---|---|---|

$\frac{1}{2}$ | $x^{1/2}=\sqrt{x}$ | $\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}$ |

$\frac{1}{3}$ | $x^{1/3}=\sqrt[3]{x}$ | $\frac{1}{3}x^{-2/3}=\frac{1}{3\sqrt[3]{x^2}}$ |

$-\frac{1}{2}$ | $x^{-1/2}=\frac{1}{\sqrt{x}}$ | $-\frac{1}{2}x^{-3/2}=-\frac{1}{2\sqrt{x^3}}$ |

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##### Math 1241, Fall 2020

##### Math 201, Spring 22

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