Here is a summary of rules of derivation (sometimes referred to as differentiation).

I wrote this post as the notation used on other sites is often ambiguous (particularly with juxtapositions that could denote either a product or a function application).

I previously wrote some brief notes on some derivations of the derivation formulae.

### Notation in the Literature

Given an expression or function *y*, there are various notations for the derivative with respect to a variable *x*.

Newton’s notations included

— in fact this was for derivation with respect to time, *t*.

Lagrange used

— though this form was really only used for higher derivatives

Leibniz used the much more suggestive notation

which had very intuitive manipulations such as

;

There is also the operator notation

and the notations for partial derivation (derivation with respect to one of many variables)

### Notation

I will use the notation for partial derivation, and write

for

To distinguish between juxtapositions and easily-confused notations, and to establish my conventions, I write

for the **product** function where

for the **composite** function where

(note that some authors prefer a reversal of this notation)

There is an ambiguity, or rather, inconsistency, with power notation for functions.

is a multiplicative product, where

whereas

is a compositional inverse, not a multiplicative reciprocal, where

However, these are so common as conventions, that I do follow them, though preferring the notation for inverse.

### Table

This table summarises some of the rules

or |
or |
---|---|

constant c |
0 |

1 | |

or | |

or |

Here are the elementary functions:

or |
or |
---|---|

exp or | exp or (again) |

ln or | or |

sin | cos |

cos | − sin |

tan | sec^{2} |

cot | − csc^{2} |

sec | |

csc | |

sinh | cosh |

cosh | sinh |

tanh | sech^{2} |

coth | − csch^{2} |

sech | |

csch | |

### Further Reading on Notation

### Further Reading for Rules

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