{"id":9297,"date":"2018-02-09T15:15:30","date_gmt":"2018-02-09T09:45:30","guid":{"rendered":"http:\/\/mycbseguide.com\/blog\/?p=9297"},"modified":"2019-03-26T12:05:50","modified_gmt":"2019-03-26T06:35:50","slug":"continuity-differentiability-class-12-notes-mathematics","status":"publish","type":"post","link":"https:\/\/mycbseguide.com\/blog\/continuity-differentiability-class-12-notes-mathematics\/","title":{"rendered":"Continuity and Differentiability class 12 Notes Mathematics"},"content":{"rendered":"<p><strong>Continuity and Differentiability Class 12 Notes Mathematics<\/strong> in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Continuity and Differentiability class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 5 Continuity and Differentiability are also available for download in CBSE Guide website.<\/p>\n<h2>Continuity and Differentiability Class 12 Notes Mathematics<\/h2>\n<p>Download CBSE class 12th revision notes for chapter 5 Continuity and Differentiability in PDF format for free. Download revision notes for Continuity and Differentiability class 12 Notes and score high in exams. These are the Continuity and Differentiability class 12 Notes prepared by team of expert teachers. The revision notes help you revise the whole chapter 5 in minutes. Revision notes in exam days is one of the best tips recommended by teachers during exam days.<\/p>\n<p style=\"text-align: center;\"><strong><a class=\"button\" href=\"https:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Download Revision Notes as PDF<\/a><\/strong><\/p>\n<h2><strong>CBSE Class 12 Mathematics Chapter 5<br \/>\nContinuity and Differentiability<br \/>\n<\/strong><\/h2>\n<ul>\n<li>Continuity of function at a point: Geometrically we say that a function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y = f\\left( x \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"64\" height=\"18\" \/><\/span><\/span>\u00a0is continuous at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"41\" height=\"7\" \/><\/span><\/span>\u00a0if the graph of the function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y = f\\left( x \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"64\" height=\"18\" \/><\/span><\/span>\u00a0is continuous (without any break) at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"41\" height=\"7\" \/><\/span><\/span>.<\/li>\n<li>A funciton <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"33\" height=\"18\" \/><\/span><\/span>\u00a0is said to be continuous at a point <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>\u00a0if:<\/li>\n<\/ul>\n<p>(i)\u00a0\u00a0\u00a0\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( a \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"33\" height=\"18\" \/><\/span><\/span>\u00a0exists i.e., <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( a \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"33\" height=\"18\" \/><\/span><\/span>\u00a0is finite, definite and real.<\/p>\n<p>(ii)\u00a0\u00a0\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a} f\\left( x \\right)\" \/><\/span><\/span>\u00a0exists.<\/p>\n<p>(iii)\u00a0\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a} f\\left( x \\right) = f\\left( a \\right)\" \/><\/span><\/span><\/p>\n<ul>\n<li>A function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right)\" \/><\/span><\/span>\u00a0is continuous at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = a\" \/><\/span><\/span>\u00a0 if <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{h \\to 0} f\\left( {a + h} \\right) = \\mathop {\\lim }\\limits_{h \\to 0} f\\left( {a + h} \\right) = f\\left( a \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"267\" height=\"25\" \/><\/span><\/span>\u00a0where <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?h \\to 0\" \/><\/span><\/span>\u00a0through positive values.<\/li>\n<li><strong>Continuity of a function in a closed interval<\/strong>: A function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right)\" \/><\/span><\/span>\u00a0is said to be continuous in the closed interval \u00a0if it is continuous for\u00a0every value of\u00a0<span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x\" \/><\/span><\/span>\u00a0lying between a and b continuous to the right of a and to the left of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = b\" \/><\/span><\/span>\u00a0i.e., <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to a - 0} f\\left( x \\right) = f\\left( a \\right)\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"138\" height=\"25\" \/><\/span><\/span>\u00a0and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\mathop {\\lim }\\limits_{x \\to b - 0} f\\left( x \\right) = f\\left( b \\right)\" \/><\/span><\/span><\/li>\n<li><strong>Continuity of a function in a open interval<\/strong>: A function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right)\" \/><\/span><\/span>\u00a0is said to be continuous in an open interval <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {a,b} \\right)\" \/><\/span><\/span>\u00a0if it is continuous at every point in\u00a0 <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {a,b} \\right)\" \/><\/span><\/span>.<\/li>\n<li><strong>Discontinuity (Discontinuous function<\/strong>): A function <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\\left( x \\right)\" \/><\/span><\/span>\u00a0is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.<\/li>\n<li>Suppose <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0is a real function and <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?c\" \/><\/span><\/span>\u00a0is a point in its domain. The derivative of <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f\" \/><\/span><\/span>\u00a0at <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?c\" \/><\/span><\/span>\u00a0is defined by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?f'\\left( c \\right) = \\mathop {\\lim }\\limits_{h \\to 0} {{f\\left( {c + h} \\right) - f\\left( c \\right)} \\over h}\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"208\" height=\"38\" \/><\/span><\/span>\u00a0provided this limit exists.<\/li>\n<li>A real valued function is continuous at a point in its domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.<\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{dy} \\over {dx}}\" \/><\/span><\/span>\u00a0is derivative of first order and is also denoted by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y'\" \/><\/span><\/span>\u00a0or <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y_1\" \/><\/span><\/span>.<\/li>\n<li>Sum, difference, product and quotient of continuous functions are continuous. i.e., if f and g are continuous functions, then <img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image001.png\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"203\" height=\"27\" \/>\u00a0is continuous. (f . g) (x) = f (x) . g(x) is continuous.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image002.png\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"100\" height=\"48\" \/>\u00a0(wherever g (x) \u2260 0) is continuous.<\/p>\n<ul>\n<li>Every differentiable function is continuous, but the converse is not true.<\/li>\n<li>Chain rule is rule to differentiate composites of functions. If f = v o u, t = u (x) and if both <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image003.png\" \/>exist then<img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image004.png\" \/><\/li>\n<li>Following are some of the standard derivatives (in appropriate domains):<\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {u \\pm v} \\right)' = u' \\pm v'\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {uv} \\right)' = u'v + uv'\" \/><\/span><\/span>\u00a0[Product Rule]<\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?\\left( {{u \\over v}} \\right)' = {{u'v - uv'} \\over {{v^2}}}\" \/><\/span><\/span>, wherever <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?v \\ne 0\" \/><\/span><\/span>\u00a0[Quotient Rule]<\/li>\n<li>If <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y = f\\left( u \\right);u = g\\left( x \\right),\" \/><\/span><\/span>\u00a0then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{dy} \\over {dx}} = {{dy} \\over {du}} \\times {{du} \\over {dx}}\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"106\" height=\"33\" \/><\/span><\/span>\u00a0[Chain Rule]<\/li>\n<li>If <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?x = f\\left( t \\right);y = g\\left( t \\right)\" \/><\/span><\/span>, then <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{dy} \\over {dx}} = {{dy} \\over {dt}} \\div {{dx} \\over {dt}}\" \/><\/span><\/span>\u00a0[Parametric Form]<\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {{x^n}} \\right) = n{x^{n - 1}}\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\sin x} \\right) = \\cos x\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\cos x} \\right) = - \\sin x\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\tan x} \\right) = - {\\sec ^2}x\" alt=\"Continuity and Differentiability class 12 Notes Mathematics\" width=\"145\" height=\"33\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\cot x} \\right) = - \\cos e{c^2}x\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\sec x} \\right) = \\sec x.\\tan x\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {\\cos ec x} \\right) = - \\cos ec x.\\cot x.\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {{a^x}} \\right) = {a^x}.{\\log _e}a\" \/><\/span><\/span><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {{{\\log }_e}x} \\right) = {1 \\over x}\" \/><\/span><\/span><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image005.png\" \/><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image006.png\" \/><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image007.png\" \/><\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{d \\over {dx}}\\left( {{{\\cos }^{ - 1}}x} \\right) = {{ - 1} \\over {1 + {x^2}}}\" \/><\/span><\/span><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image009.png\" \/><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image010.png\" \/><\/li>\n<li><img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image011.png\" \/><\/li>\n<li>Logarithmic differentiation is a powerful technique to differentiate functions of the form <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image013.png\" \/>\u00a0\u00a0\u00a0Here both f (x) and u (x) need to be positive for this technique to make sense.<\/li>\n<li>If we have to differentiate logarithmic funcitons, other than base <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?e\" \/><\/span><\/span>, then we use the result <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{\\log _b}a = {{{{\\log }_e}a} \\over {{{\\log }_e}b}}\" \/><\/span><\/span>\u00a0and then differentiate R.H.S.<\/li>\n<li>While differentiating inverse trigonometric functions, first represent it in simplest form by using suitable substitution and then differentiate simplified form.<\/li>\n<li>If we are given implicit functions then differentiate directly w.r.t. suitable variable involved and get the derivative by readusting the terms.<\/li>\n<li><span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?{{{d^2}y} \\over {d{x^2}}} = {d \\over {dx}}\\left( {{{dy} \\over {dx}}} \\right)\" \/><\/span><\/span>\u00a0is derivative of second order and is denoted by <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y''\" \/><\/span><\/span>\u00a0or <span class=\"cke_widget_wrapper cke_widget_inline cke_widget_selected\"><span class=\"math-tex cke_widget_element\"><img decoding=\"async\" src=\"https:\/\/elpiscart.com\/cgi-bin\/mathtex.cgi?y_2\" \/><\/span><\/span>.<\/li>\n<li><strong>Rolle\u2019s Theorem: <\/strong>If f: [a, b] \u2192 R is continuous on [a, b] and differentiable on (a, b) such that f (a) = f (b), then there exists some c in (a, b) such that f \u2032(c) = 0.<\/li>\n<li><strong>Lagrange&#8217;s Mean Value Theorem<\/strong>: If f: [a, b] \u2192 R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that <img decoding=\"async\" src=\"https:\/\/media-mycbseguide.s3.amazonaws.com\/images\/static\/revise\/12\/mathematics\/ch05\/image014.png\" \/><\/li>\n<\/ul>\n<h2><strong>CBSE Class 12 Revision Notes and Key Points<\/strong><\/h2>\n<p>Continuity and Differentiability class 12 Notes Mathematics. CBSE quick revision note for class-12 Chemistry Physics Math\u2019s, Biology and other subject are very helpful to revise the whole syllabus during exam days. The revision notes covers all important formulas and concepts given in the chapter. Even if you wish to have an overview of a chapter, quick revision notes are here to do if for you. These notes will certainly save your time during stressful exam days.<\/p>\n<ul>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physics\/1251\/cbse-revision-notes\/7\/\">Physics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-chemistry\/1267\/cbse-revision-notes\/7\/\">Chemistry<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-mathematics\/1284\/cbse-revision-notes\/7\/\">Mathematics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-biology\/1298\/cbse-revision-notes\/7\/\">Biology<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-accountancy\/1315\/cbse-revision-notes\/7\/\">Accountancy<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-economics\/1327\/cbse-revision-notes\/7\/\">Economics<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-business-studies\/1727\/cbse-revision-notes\/7\/\">Business Studies<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-computer-science\/1851\/cbse-revision-notes\/7\/\">Computer Science<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-informatics-practices\/1873\/cbse-revision-notes\/7\/\">Informatics Practices<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-english-core\/1855\/cbse-revision-notes\/7\/\">English Core<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-history\/1869\/cbse-revision-notes\/7\/\">History<\/a><\/li>\n<li><a href=\"http:\/\/mycbseguide.com\/downloads\/cbse-class-12-physical-education\/1877\/cbse-revision-notes\/7\/\">Physical Education<\/a><\/li>\n<\/ul>\n<p>To download Continuity and Differentiability class 12 Notes, sample paper for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise pract, ce, NCERT Continuity and Differentiability, NCERT Exemplar Continuity and Differentiability, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through\u00a0<a href=\"https:\/\/play.google.com\/store\/apps\/details?id=in.techchefs.MyCBSEGuide&amp;referrer=utm_source%3Dmycbse_bottom%26utm_medium%3Dtext%26utm_campaign%3Dmycbseads\"><strong>the best app for CBSE students<\/strong><\/a>\u00a0and myCBSEguide website.<\/p>\n<ul>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/relations-functions-class-12-notes-mathematics\/\">Relations and Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/inverse-trigonometric-functions-class-12-notes-mathematics\/\">Inverse Trigonometric Functions class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/matrices-class-12-notes-mathematics\/\">Matrices class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/determinants-class-12-notes-mathematics\/\">Determinants class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-derivatives-class-12-notes-mathematics\/\">Application of Derivatives class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/integrals-class-12-notes-mathematics\/\">Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/application-integrals-class-12-notes-mathematics\/\">Application of Integrals class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/differential-equations-class-12-notes-mathematics\/\">Differential Equations class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/vector-algebra-class-12-notes-mathematics\/\">Vector Algebra class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/three-dimensional-geometry-class-12-notes-mathematics\/\">Three Dimensional Geometry class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/linear-programming-class-12-notes-mathematics\/\">Linear Programming class 12 Notes Mathematics<\/a><\/li>\n<li class=\"entry-title\"><a href=\"https:\/\/mycbseguide.com\/blog\/probability-class-12-notes-mathematics\/\">Probability class 12 Notes Mathematics<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Continuity and Differentiability Class 12 Notes Mathematics in PDF are available for free download in myCBSEguide mobile app. The best app for CBSE students now provides Continuity and Differentiability class 12 Notes latest chapter wise notes for quick preparation of CBSE board exams and school-based annual examinations. Class 12 Mathematics notes on chapter 5 Continuity &#8230; <a title=\"Continuity and Differentiability class 12 Notes Mathematics\" class=\"read-more\" href=\"https:\/\/mycbseguide.com\/blog\/continuity-differentiability-class-12-notes-mathematics\/\" aria-label=\"More on Continuity and Differentiability class 12 Notes Mathematics\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":9266,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[48,456],"tags":[457,150,559,461,426,240],"class_list":["post-9297","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-cbse-class-12","category-revision-notes","tag-cbse-notes","tag-cbse-notes-and-key-points","tag-continuity-and-differentiability","tag-mathematics-notes","tag-quick-revision","tag-quick-revision-notes"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - 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