https://calculus.subwiki.org/w/index.php?title=Discrete_derivative&feed=atom&action=history Discrete derivative - Revision history 2020-10-28T11:52:39Z Revision history for this page on the wiki MediaWiki 1.29.2 https://calculus.subwiki.org/w/index.php?title=Discrete_derivative&diff=1778&oldid=prev Vipul: Created page with "==Definition== The term '''discrete derivative''' is a loosely used term to describe an analogue of derivative for a function whose domain is discrete. The idea is typica..." 2012-06-04T17:46:44Z <p>Created page with &quot;==Definition== The term &#039;&#039;&#039;discrete derivative&#039;&#039;&#039; is a loosely used term to describe an analogue of <a href="/wiki/Derivative" title="Derivative">derivative</a> for a function whose domain is discrete. The idea is typica...&quot;</p> <p><b>New page</b></p><div>==Definition==<br /> <br /> The term '''discrete derivative''' is a loosely used term to describe an analogue of [[derivative]] for a function whose domain is discrete. The idea is typically to define this as a [[difference quotient]] rather than the usual ''continuous'' notion of derivative, which is defined as a limit of a difference quotient.<br /> <br /> The typical case of interest is a function defined on the set of integers, or some contiguous subset of the set of integers (for instance, all integers from &lt;math&gt;a&lt;/math&gt; to &lt;math&gt;b&lt;/math&gt;, where &lt;math&gt;a &lt; b&lt;/math&gt; are integers). There are two related notions:<br /> <br /> * The '''forward difference operator''', sometimes denoted &lt;matH&gt;\Delta&lt;/math&gt;, is defined as follows for a function &lt;math&gt;f&lt;/math&gt;:<br /> <br /> &lt;math&gt;\Delta f = n \mapsto f(n + 1) - f(n)&lt;/math&gt;<br /> <br /> This can be thought as a [[difference quotient]] between &lt;math&gt;n&lt;/math&gt; and &lt;matH&gt;n + 1&lt;/math&gt;. Note that it is analogous to the right hand derivative.<br /> <br /> * The '''backward difference operator''' is defined as:<br /> <br /> &lt;math&gt;n \mapsto f(n) - f(n - 1)&lt;/math&gt;<br /> <br /> This can be thought as a [[difference quotient]] between &lt;math&gt;n&lt;/math&gt; and &lt;matH&gt;n - 1&lt;/math&gt;. Note that it is analogous to the left hand derivative.<br /> <br /> In practice, we simply choose one of these as the notion of discrete derivative and stick with it. The reason is that the forward difference operator of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;n&lt;/math&gt; equals the backward difference operator of &lt;math&gt;f&lt;/math&gt; at &lt;math&gt;n + 1&lt;/math&gt;, so we do not in fact lose any information by considering only one of these operators as the discrete derivative.</div> Vipul