{"id":31,"date":"2004-06-06T14:25:10","date_gmt":"2004-06-06T14:25:10","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=31"},"modified":"2007-10-31T14:51:50","modified_gmt":"2007-10-31T14:51:50","slug":"generalisation-of-derivative","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=31","title":{"rendered":"Generalisation of derivative"},"content":{"rendered":"<p><font size=1><i>Inspired by a posting on <a href=\"http:\/\/www.sosmath.com\/CBB\/index.php\">S.O.S. Mathematics CyberBoard<\/a> <\/i><\/font><\/p>\n<p>Most students will be familiar with the definition of the derivative of a real-valued function of a real variable defined on some interval (a,b):<\/p>\n<ul>If <img src='\/maths\/latexrender\/pictures\/d9a228a685dd5c1f9f4c32e048440c59.gif' title='x \\in (a,b)' alt='x \\in (a,b)' align=absmiddle> then f is differentiable at <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle> if <img src='\/maths\/latexrender\/pictures\/3f717b3dce2026026928ac8a18da3315.gif' title='\\displaystyle \\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}' alt='\\displaystyle \\lim_{h \\to 0}\\frac{f(x+h)-f(x)}{h}' align=absmiddle> exists and the limit is denoted <img src='\/maths\/latexrender\/pictures\/7dbc3175c8cebe624521610ef4c067cc.gif' title='f^{\\prime}(x)' alt='f^{\\prime}(x)' align=absmiddle><\/ul>\n<p>It is also clear that for this to make sense <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> must be defined at <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle> (and of course it is a well-known consequence of the definition that <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is also continuous at <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle>). But what if <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is defined on <img src='\/maths\/latexrender\/pictures\/2d05e1f15387f87456155cd96cc06235.gif' title='(a,b)' alt='(a,b)' align=absmiddle> but not at <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle>, can we do anything then? Yes, we can define a pseudo-derivative <img src='\/maths\/latexrender\/pictures\/ec460f552938e0ba4716b42fb4937a32.gif' title='\\Lambda f' alt='\\Lambda f' align=absmiddle> of <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> provided <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is defined on a neighbourhood of <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle>:<\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/14efbdac0315a10a9246f4b2a6948c7d.gif' title='\\displaystyle \\Lambda f(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x-h)}{2h}' alt='\\displaystyle \\Lambda f(x)=\\lim_{h \\to 0}\\frac{f(x+h)-f(x-h)}{2h}' align=absmiddle><\/ul>\n<p>This pseudo-derivative has similar properties to the derivative and indeed it has the same values where <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is differentiable but there are significant differences as the following exercises show:<\/p>\n<ol Type=1>\n<li> If <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is differentiable at <img src='\/maths\/latexrender\/pictures\/9dd4e461268c8034f5c8564e155c67a6.gif' title='x' alt='x' align=absmiddle> show that <img src='\/maths\/latexrender\/pictures\/f88721fcf44e5dbc71e2b5f630b4313a.gif' title='\\Lambda f(x)=f^{\\prime}x)' alt='\\Lambda f(x)=f^{\\prime}x)' align=absmiddle><\/li>\n<p><\/p>\n<li> If <img src='\/maths\/latexrender\/pictures\/41d8ce86d818b46ed767d77d1f2868bc.gif' title='f(x)=|x|' alt='f(x)=|x|' align=absmiddle> show that <img src='\/maths\/latexrender\/pictures\/e5caeff07c639ed756238f5ba442dbc8.gif' title='\\Lambda f(0)' alt='\\Lambda f(0)' align=absmiddle> exists although <img src='\/maths\/latexrender\/pictures\/6d86a78e3371f77f588e2877f411b576.gif' title='f^{\\prime}(0)' alt='f^{\\prime}(0)' align=absmiddle> does not<\/li>\n<p><\/p>\n<li> If <img src='\/maths\/latexrender\/pictures\/b3a090f422a1fd3907fb17e0caedb2ac.gif' title='f(x)=\\left\\{\\begin{array}{ll}x &amp; \\mbox{ if } x&amp;lt;0\\\\-x^2 &amp; \\mbox{ if } x \\geq 0\\end{array}\\right.' alt='f(x)=\\left\\{\\begin{array}{ll}x &amp; \\mbox{ if } x&amp;lt;0\\\\-x^2 &amp; \\mbox{ if } x \\geq 0\\end{array}\\right.' align=absmiddle> show that <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> has a local maximum at 0 but <img src='\/maths\/latexrender\/pictures\/4acfef45c3811a2c366211eacab9db7c.gif' title='\\Lambda f(0) \\neq 0' alt='\\Lambda f(0) \\neq 0' align=absmiddle><\/li>\n<p><\/p>\n<li> Suppose <img src='\/maths\/latexrender\/pictures\/8fa14cdd754f91cc6554c9e71929cce7.gif' title='f' alt='f' align=absmiddle> is differentiable on <img src='\/maths\/latexrender\/pictures\/2d05e1f15387f87456155cd96cc06235.gif' title='(a,b)' alt='(a,b)' align=absmiddle>, except at a point <img src='\/maths\/latexrender\/pictures\/3e0d691f3a530e6c7e079636f20c111b.gif' title='x_0' alt='x_0' align=absmiddle> in <img src='\/maths\/latexrender\/pictures\/2d05e1f15387f87456155cd96cc06235.gif' title='(a,b)' alt='(a,b)' align=absmiddle>, with <img src='\/maths\/latexrender\/pictures\/c67a4aee10137dcb695268d87b5b546f.gif' title='f^{\\prime}(x)\\geq 0' alt='f^{\\prime}(x)\\geq 0' align=absmiddle> for <img src='\/maths\/latexrender\/pictures\/b1029cd1b01f67f866334020b5ff7403.gif' title=' x \\neq x_0' alt=' x \\neq x_0' align=absmiddle>.<br \/>\nIf <img src='\/maths\/latexrender\/pictures\/87730b01f8290e1745f96518fd6d2516.gif' title='\\Lambda (f)(x_0) ' alt='\\Lambda (f)(x_0) ' align=absmiddle> exists show that <img src='\/maths\/latexrender\/pictures\/864c31f0cbf28bc74750351707aaac9c.gif' title='\\Lambda (f)(x)\\geq0' alt='\\Lambda (f)(x)\\geq0' align=absmiddle><\/li>\n<\/ol>\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inspired by a posting on S.O.S. Mathematics CyberBoard Most students will be familiar with the definition of the derivative of a real-valued function of a real variable defined on some interval (a,b): If then f is differentiable at if exists and the limit is denoted It is also clear that for this to make sense [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-31","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/31","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=31"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/31\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=31"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=31"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=31"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}