{"id":694,"date":"2024-04-20T10:00:08","date_gmt":"2024-04-20T10:00:08","guid":{"rendered":"https:\/\/gbelentzas.sites.sch.gr\/?p=694"},"modified":"2024-05-01T07:42:05","modified_gmt":"2024-05-01T07:42:05","slug":"694","status":"publish","type":"post","link":"https:\/\/gbelentzas.sites.sch.gr\/?p=694","title":{"rendered":"\u03a0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 ln|x| (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7)"},"content":{"rendered":"

\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 15. \u00a0<\/strong>\u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 <\/span>\"\textcolor{blue}{f(x)=\ln |x|, x \in \mathbf{R}^* }\". \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7<\/span> \"f\" \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf <\/span>\"\textcolor{blue}{\rr^*}\" \u00a0\u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 <\/span>\"\textcolor{blue}{f^{\prime}(x)=\dfrac{1}{x} }\" \u00a0\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae<\/span><\/p>\n

\n
\n

  <\/span>   <\/span>\"\[\textcolor{blue}{(\ln |x|)^{\prime}=\dfrac{1}{x} }\]\"<\/p>\n<\/div>\n<\/div>\n

(\u0397\u039c. 2008)<\/em><\/span><\/p>\n

\n

\u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \"f(x)=\ln |x|, x \in \mathbf{R}^*\"<\/p>\n

    \n
  • \u0391\u03bd \"x > 0,\" \u03c4\u03cc\u03c4\u03b5 \"(\ln |x|)^{\prime}=(\ln x)^{\prime}=\dfrac{1}{x}\"<\/li>\n
  • \u0391\u03bd \"x < 0,\" \u03c4\u03cc\u03c4\u03b5 \"\ln |x| = \ln (−x),\" \u03bf\u03c0\u03cc\u03c4\u03b5, \u03b1\u03bd \u03b8\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \\begin{center}<\/li>\n<\/ul>\n

    \"y = ln(−x)\" \u03ba\u03b1\u03b9 \"u = −x,\"\\end{center}<\/p>\n

    \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 y = lnu. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2,<\/p>\n

    \n

      <\/span>   <\/span>\"\[y^{\prime}=(\ln u)^{\prime}=\dfrac{1}{u} \cdot u^{\prime}=\dfrac{1}{-x}(-1)=\dfrac{1}{x}\]\"<\/p>\n<\/p>\n

    \u0386\u03c1\u03b1, <\/p>\n

      <\/span>   <\/span>\"\[(\ln |x|)^{\prime}=\dfrac{1}{x}\]\"<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 15. \u00a0\u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 . \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf \u00a0\u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u00a0\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae     (\u0397\u039c. 2008) \u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u0391\u03bd \u03c4\u03cc\u03c4\u03b5 \u0391\u03bd \u03c4\u03cc\u03c4\u03b5 \u03bf\u03c0\u03cc\u03c4\u03b5, \u03b1\u03bd \u03b8\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \\begin{center} \u03ba\u03b1\u03b9 \\end{center} \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 y = lnu. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2,     \u0386\u03c1\u03b1,    <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-694","post","type-post","status-publish","format-standard","hentry","category-13"],"featured_image_src":null,"author_info":{"display_name":"admin","author_link":"https:\/\/gbelentzas.sites.sch.gr\/?author=1"},"yoast_head":"\n\u03a0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 ln|x| (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7) - MathsEdu.gr<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/gbelentzas.sites.sch.gr\/?p=694\" \/>\n<meta property=\"og:locale\" content=\"el_GR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u03a0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 ln|x| (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7) - MathsEdu.gr\" \/>\n<meta property=\"og:description\" content=\"\u0398\u03b5\u03ce\u03c1\u03b7\u03bc\u03b1 15. \u00a0\u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 . \u039d\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03af\u03be\u03b5\u03c4\u03b5 \u03cc\u03c4\u03b9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03af\u03c3\u03b9\u03bc\u03b7 \u03c3\u03c4\u03bf \u00a0\u03ba\u03b1\u03b9 \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u00a0\u03b4\u03b7\u03bb\u03b1\u03b4\u03ae     (\u0397\u039c. 2008) \u0388\u03c3\u03c4\u03c9 \u03b7 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u0391\u03bd \u03c4\u03cc\u03c4\u03b5 \u0391\u03bd \u03c4\u03cc\u03c4\u03b5 \u03bf\u03c0\u03cc\u03c4\u03b5, \u03b1\u03bd \u03b8\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 begin{center} \u03ba\u03b1\u03b9 end{center} \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 y = lnu. \u0395\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2,     \u0386\u03c1\u03b1,    \" \/>\n<meta property=\"og:url\" content=\"https:\/\/gbelentzas.sites.sch.gr\/?p=694\" \/>\n<meta property=\"og:site_name\" content=\"MathsEdu.gr\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/MathsEdu.gr\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-20T10:00:08+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-01T07:42:05+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"\u03a3\u03c5\u03bd\u03c4\u03ac\u03c7\u03b8\u03b7\u03ba\u03b5 \u03b1\u03c0\u03cc\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694\"},\"author\":{\"name\":\"admin\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/#\\\/schema\\\/person\\\/511115b6820d1a71ade6791721485e15\"},\"headline\":\"\u03a0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 ln|x| (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7)\",\"datePublished\":\"2024-04-20T10:00:08+00:00\",\"dateModified\":\"2024-05-01T07:42:05+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694\"},\"wordCount\":86,\"commentCount\":0,\"articleSection\":[\"\u0391\u03a0\u039f\u0394\u0395\u0399\u039e\u0395\u0399\u03a3\"],\"inLanguage\":\"el\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694\",\"url\":\"https:\\\/\\\/gbelentzas.sites.sch.gr\\\/?p=694\",\"name\":\"\u03a0\u03b1\u03c1\u03ac\u03b3\u03c9\u03b3\u03bf\u03c2 \u03c4\u03b7\u03c2 ln|x| (\u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7) - 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