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.
— 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)
I will use the notation for partial derivation, and write
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
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.
This table summarises some of the rules
Here are the elementary functions:
|exp or||exp or (again)|
Further Reading on Notation
Further Reading for Rules