The derivative of a Function represents an infinitesimal change in the function with respect to whatever parameters it
may have. The ``simple'' derivative of a function with respect to is denoted either or (and
often written in-line as ). When derivatives are taken with respect to time, they are often denoted using
Newton's Fluxion notation,
. The derivative of a function with
respect to the variable is defined as

(1) |

A 3-D generalization of the derivative to an arbitrary direction is known as the Directional Derivative. In general, derivatives are mathematical objects which exist between smooth functions on manifolds. In this formalism, derivatives are usually assembled into ``Tangent Maps.''

Simple derivatives of some simple functions follow.

(2) | |

(3) | |

(4) | |

(5) | |

(6) | |

(7) | |

(8) | |

(9) | |

(10) | |

(11) | |

(12) | |

(13) | |

(14) | |

(15) | |

(16) | |

(17) | |

(18) | |

(19) | |

(20) | |

(21) | |

(22) | |

(23) | |

(24) | |

(25) | |

(26) |

Derivatives of sums are equal to the sum of derivatives so that

(27) |

(28) |

(29) |

(30) |

Other rules involving derivatives include the Chain Rule, Power Rule, Product Rule, and Quotient Rule. Miscellaneous other derivative identities include

(31) |

(32) |

(33) |

(34) |

A vector derivative of a vector function

(35) |

(36) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 11, 1972.

Anton, H. *Calculus with Analytic Geometry, 5th ed.* New York: Wiley, 1987.

Beyer, W. H. ``Derivatives.'' *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL:
CRC Press, pp. 229-232, 1987.

© 1996-9

1999-05-24