{"id":5887,"date":"2025-08-11T10:07:24","date_gmt":"2025-08-11T04:37:24","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=5887"},"modified":"2025-08-11T10:17:40","modified_gmt":"2025-08-11T04:47:40","slug":"differentiability","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/calculus-and-differeential-equations\/differentiability","title":{"rendered":"Differentiability"},"content":{"rendered":"<p>Derivative at a point: Let I denote the open interval (a, b) in R, and let x\u2080 \u2208 I. Then a function f: I \u2192 R is said to be differentiable at x0, if<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5888 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/08\/Differentiability.jpg\" alt=\"Differentiability\" width=\"309\" height=\"58\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/Differentiability.jpg 309w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/Differentiability-300x56.jpg 300w\" sizes=\"auto, (max-width: 309px) 100vw, 309px\" \/><\/p>\n<p>exist (finitely) and is denoted by f\u2032(x0).<\/p>\n<h2>Differentiability in an Open Interval<\/h2>\n<p>A function f is said to be differentiable in an open interval (a, b), if it is differentiable at each point of the open interval.<\/p>\n<h3>Differentiability in a Closed Interval<\/h3>\n<p>A function f : [a, b] \u2192 R is said to be differentiable in closed interval [a, b] if it is<\/p>\n<p><strong>1.<\/strong> differentiable from right a t a [i.e. R f\u2032(a) exists] and<br \/>\n<strong>2.<\/strong> differentiable from left at b [i.e. L f\u2032(a) exists] and<br \/>\n<strong>3.<\/strong> differentiable in the open interval (a, b).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Derivative at a point: Let I denote the open interval (a, b) in R, and let x\u2080 \u2208 I. Then<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1733,5],"tags":[1738,1740],"class_list":["post-5887","post","type-post","status-publish","format-standard","hentry","category-calculus-and-differeential-equations","category-ee","tag-differentiability","tag-differentiability-in-a-closed-interval"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5887","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=5887"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5887\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=5887"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=5887"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=5887"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}