{"id":23351,"date":"2021-11-13T20:15:47","date_gmt":"2021-11-13T14:45:47","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=23351"},"modified":"2024-07-09T19:41:47","modified_gmt":"2024-07-09T14:11:47","slug":"differentiation","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=23351","title":{"rendered":"DIFFERENTIATION (Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[By\\ knowing\\ the\\ position\\ of\\ a\\ body\\ at\\ various\\ time\\ intervals\\ it\\ is\\ possible\\ to\\ find\\ the\\ rate\\ at\\ which\\ the\\ position\\ of\\ the\\ body\\ is\\]\\[ changing.\\ It\\ is\\ of \\ very\\ general\\ interest\\ to\\ know\\ a\\ certain\\ parameter\\ at\\ various\\ instants\\ of\\ time\\ and\\ try\\ to\\ finding\\ the\\ rate\\]\\[ at\\ which\\ it\\ is\\ changing.\\ There\\ are\\ several\\ real\\ life\\ situations\\ where\\ such\\ a\\ process\\ needs\\ to\\ be\\ carried\\ out.\\ For\\ instance,\\ people\\] \\[maintaining\\ a\\ reservoir\\ need\\ to\\ know\\ when\\ will\\ a\\ reservoir\\ overflow\\ knowing\\ the\\ depth\\ of\\ the\\ water\\ at\\ several\\ instances\\ of\\ time,\\]\\[ Rocket\\ Scientists\\ need\\ to\\ compute\\ the\\ precise\\ velocity\\ with\\ which\\ the\\ satellite\\ needs\\ to\\ be\\ shot\\ out\\ from\\ the\\ rocket\\] \\[knowing\\ the\\ height\\ of\\ the\\ rocket\\ at\\ various\\ times.\\ Financial\\ institutions\\ need\\ to\\ predict\\ the\\ changes\\ in\\ the\\ value\\ of\\ a\\ particular\\] \\[stock\\ knowing\\ its\\ present\\ value.\\ In\\ these,\\ and\\ many\\ such\\ cases\\ it\\ is\\ desirable\\ to\\ know\\ how\\ a\\ particular\\ parameter\\ is\\ changing\\]\\[ with\\ respect\\ to\\ some\\ other\\ parameter.\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Suppose\\  f\\  is\\  a\\  real\\ valued\\ function,\\  the\\ function\\ defined\\ by\\  \\lim\\ _{h\\ \\to\\ 0}\\ \\frac{f(x\\ +\\ h)\\ -\\ f(x)}{x\\ -\\ h}\\]\\[is\\ defined\\ to\\ be\\ the\\ derivative\\ of\\ f\\ at\\ x\\ and\\ is\\ denoted\\ by\\ f\\prime(x).\\]\\[f\\prime(x)\\ is\\ denoted\\ by\\ \\frac{d}{dx}\\ (f(x))\\ or\\ \\frac{dy}{dx}\\ (where\\ y\\ =\\ f(x))\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:160px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Formulae}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\frac{d}{dx}\\ (x^n)\\ =\\ n\\ x^{n\\ -\\ 1}\\]\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\frac{d}{dx}\\ (\\sqrt{x})\\ =\\ \\frac{1}{2\\ \\sqrt{x}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\frac{d}{dx}\\ (e^x)\\ =\\ e^x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\frac{d}{dx}\\ (log\\ x)\\ =\\ \\frac{1}{x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\frac{d}{dx}\\ (Sin\\ x)\\ =\\ Cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[6.\\ \\frac{d}{dx}\\ (Cos\\ x)\\ =\\ -\\ Sin\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\ \\frac{d}{dx}\\ (Tan\\ x)\\ =\\ Sec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\frac{d}{dx}\\ (Cot\\ x)\\ =\\ -\\ Cosec^2\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[9.\\ \\frac{d}{dx}\\ (Sec\\ x)\\ =\\ Sec\\ x\\ Tan\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[10.\\ \\frac{d}{dx}\\ (Cosec\\ x)\\ =\\ -\\ Cosec\\ x\\ Cot\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\frac{d}{dx}\\ (a^x)\\ =\\ a^x\\ log\\ a\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\frac{d}{dx}\\ (Sin^{-1}\\ x)\\ =\\ \\frac{1}{\\sqrt{1\\ -\\ x^2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\frac{d}{dx}\\ (Cos^{-1}\\ x)\\ =\\ -\\ \\frac{1}{\\sqrt{1\\ -\\ x^2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\frac{d}{dx}\\ (Tan^{-1}\\ x)\\ =\\ \\frac{1}{1\\ +\\ x^2}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Properties}:\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ If\\ u\\ and\\ v\\ are\\ functions\\ of\\ x,\\ Then\\  \\frac{d}{dx}\\ (u\\ \\pm\\ v)\\ =\\  \\frac{du}{dx}\\ \\pm\\ \\frac{dv}{dx}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ If\\ u\\ is\\ a\\ function\\ of\\ x,\\ and\\ k\\ is\\ a\\ constant,\\  Then\\  \\frac{d}{dx}\\ (ku)\\ =\\ k \\frac{du}{dx}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ If\\ k\\ is\\ any\\ constant,\\  Then\\  \\frac{d}{dx}\\ (k)\\ =\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ If\\ u\\ and\\ v\\ are\\ functions\\ of\\ x,\\ Then\\  \\frac{d}{dx}\\ (u\\ v)\\ =\\  u\\ \\frac{dv}{dx}\\ +\\ v\\ \\frac{du}{dx}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ If\\ u\\  v\\ and\\ w\\ are\\ functions\\ of\\ x,\\ Then\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ If\\ u\\ and\\ v\\ are\\ functions\\ of\\ x,\\ Then\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\ y\\ =\\ x\\ +\\ x^2\\ +\\ x^3\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ x\\ +\\ x^2\\ +\\ x^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ \\frac{d}{dx}(x)\\ +\\ \\frac{d}{dx}(x^2)\\ +\\ \\frac{d}{dx}(x^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 1\\ +\\ 2\\ x\\ +\\ 3\\ x^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ \\color {Red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\ 3\\  x^2\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ 3\\  x^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\  3\\ \\frac{d}{dx}(x^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 3\\ (2x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 6\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\ 3\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ \\frac{d}{dx}(3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\  x^2\\ +\\ x^3\\ +\\ Cos\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ x^2\\ +\\ x^3\\ +\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ \\frac{d}{dx}(x^2)\\ +\\ \\frac{d}{dx}(x^3)\\ +\\ \\frac{d}{dx}(Cos\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 2\\ x\\ +\\ 3\\ x^2\\ -\\ Sin\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\ y\\ =\\ \\frac{1}{x^2} +\\ 3\\ tan\\ x\\ -\\ log\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ Feb\\ 2022,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ \\frac{1}{x^2} +\\ 3\\ tan\\ x\\  -\\  log\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ \\frac{d}{dx}(x^{-2})\\ +\\ 3\\ \\frac{d}{dx}(tan\\ x)\\ -\\  \\frac{d}{dx}(log\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ (-\\ 2)\\ x^{-2\\ -\\ 1}\\ +\\ 3\\  Sec^2\\ x\\ -\\ \\frac{1}{x}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ -\\ 2\\  x^{-3}\\ +\\ 3\\  Sec^2\\ x\\ -\\ \\frac{1}{x}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\  \\frac{5}{x^2}\\ +\\ \\frac{2}{x}\\ +\\ \\frac{3}{Cos\\ x}\\ +\\ \\frac{1}{8}  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ \\frac{5}{x^2}\\ +\\ \\frac{2}{x}\\ +\\ \\frac{3}{Cos\\ x}\\ +\\ \\frac{1}{8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 5\\ \\frac{d}{dx}(x^{-2})\\ +\\ 2\\ \\frac{d}{dx}(x^{-1})\\ +\\ 3\\ \\frac{d}{dx}(Sec\\ x)\\ +\\ \\frac{d}{dx}(\\frac{1}{8})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 5\\ (-\\ 2\\ x^{-3})\\ +\\ 2\\ (-\\ x^{-2})\\ +\\ 3\\ Sec\\ x\\ Tam\\ x\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ -\\ 10\\ x^{-3}\\ -\\ 2\\  x^{-2}\\ +\\ 3\\ Sec\\ x\\ Tam\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\ e^x\\ log\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ e^x\\ log\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{5cm}\\ v\\ =\\ log\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v)\\ =\\  u\\ \\frac{dv}{dx}\\ +\\ v\\ \\frac{du}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ e^x\\ \\frac{d}{dx}(log\\ x)\\ +\\ log\\ x\\ \\frac{d}{dx}(e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ e^x\\ \\frac{1}{x}\\ +\\ log\\ x\\ e^x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\ e^x\\ sin\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ e^x\\ sin\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{5cm}\\ v\\ =\\ sin\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v)\\ =\\  u\\ \\frac{dv}{dx}\\ +\\ v\\ \\frac{du}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ e^x\\ \\frac{d}{dx}(sin\\ x)\\ +\\ sin\\ x\\ \\frac{d}{dx}(e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ e^x\\ cos\\ x\\ +\\ sin\\ x\\ e^x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 9:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\  y\\ =\\ (x\\ +\\ 2)\\ (x\\ -\\ 3)\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ (x\\ +\\ 2)\\ (x\\ -\\ 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x\\ +\\ 2),\\ \\hspace{5cm}\\ v\\ =\\ (x\\ -\\ 3)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v)\\ =\\  u\\ \\frac{dv}{dx}\\ +\\ v\\ \\frac{du}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ (x\\ +\\ 2)\\ \\frac{d}{dx}(x\\ -\\ 3)\\ +\\ (x\\ -\\ 3)\\ \\frac{d}{dx}(x\\ +\\ 2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ (x\\ +\\ 2)\\ (1)\\ +\\ (x\\ -\\ 3)\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x\\ +\\ 2\\ +\\ x\\ -\\ 3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 2\\ x\\ -\\ 1\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ if\\ y\\ =\\ \\frac{1\\ +\\ cos\\ x}{1\\ -\\ cos\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ Feb\\ 2022\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ \\frac{1\\ +\\ cos\\ x}{1\\ -\\ cos\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ 1\\ +\\ cos\\ x,\\ \\hspace{5cm}\\ v\\ =\\ 1\\ -\\ cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(1\\ -\\ cos\\ x)\\ \\frac{d}{dx}\\ (1\\ +\\ cos\\ x)\\  -\\ (1\\ +\\ cos\\ x)\\ \\frac{d}{dx}\\ (1\\ -\\ cos\\ x)}{(1\\ -\\ cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(1\\ -\\ cos\\ x)\\ (-\\ Sin\\ x)\\  -\\ (1\\ +\\ cos\\ x)\\  (Sin\\ x)}{(1\\ -\\ cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{-\\ Sin\\ x)\\  +\\ Sin\\ x\\ Cos\\ x\\ -\\  Sin\\ x\\ -\\ Sin\\ x\\ Cos\\ x}{(1\\ -\\ cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{-\\ 2\\ Sin\\ x}{(1\\ -\\ cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\ \\color {red} {Differentiate\\ the\\ following\\ with\\ respect\\ to\\ x}\\ (i)\\ if\\  y\\ =\\ x^3\\ (1\\ +\\ log\\ x)\\ (ii)\\ y\\ =\\ \\frac{x +\\ Tan\\ x}{Cos\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\   (i)\\  y\\ =\\ x^3\\ (1\\ +\\ log\\ x)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^3,\\ \\hspace{5cm}\\ v\\ =\\ (1\\ +\\ log\\ x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v)\\ =\\  u\\ \\frac{dv}{dx}\\ +\\ v\\ \\frac{du}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^3\\ \\frac{d}{dx}(1\\ +\\ log\\ x)\\ +\\ (1\\ +\\ Log\\ x)\\ \\frac{d}{dx}(x^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^3\\ (0\\ +\\ \\frac{1}{x})\\ +\\ (1\\ +\\ log\\ x)\\ (3\\ x^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^2\\  +\\ 3\\ x^2\\ (1\\ +\\ log\\ x)\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{x\\ +\\ Tan\\ x}{Cos\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x\\ +\\ Tan\\ x),\\ \\hspace{5cm}\\ v\\ =\\ Cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{Cos\\ x\\ \\frac{d}{dx}\\ (x\\ +\\ Tan\\ x)\\  -\\ (x\\ +\\ Tan\\ x)\\ \\frac{d}{dx}\\ (Cos\\ x)}{(Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{Cos\\ x\\ (1\\ +\\ Sec^2\\ x)\\  -\\ (x\\ +\\ Tan\\ x)\\  (-Sin\\ x)}{(Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/PtTW7fZ7QzU\" title=\"Differentiation - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\" \"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ \\color {red} {Differentiate\\ the\\ following\\ with\\ respect\\ to\\ x}\\ (i)\\ if\\  y\\ =\\ x^3\\ Sin\\ x\\ Tan\\ x\\ (ii)\\ y\\ =\\ \\frac{x +\\ 6}{x\\ -\\ 7}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\   (i)\\  y\\ =\\ x^3\\ Sin\\ x\\ Tan\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^3,\\ \\hspace{2cm}\\ v\\ =\\ Sin\\ x\\ \\hspace{2cm}\\ w\\ =\\ Tan\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  x^3\\ Sin\\ x\\ \\frac{d}{dx}\\ (Tan\\ x)\\ +\\ Sin\\ x\\ Tan\\ x\\ \\frac{d}{dx}\\ (x^3)\\  +\\ Tan\\ x\\ x^3\\ \\frac{d}{dx}\\ (Sin\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  x^3\\ Sin\\ x\\ (Sec^2\\ x)\\ +\\ Sin\\ x\\ Tan\\ x\\ 3\\ x^2\\  +\\ Tan\\ x\\ x^3\\ Cos\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  x^3\\ Sin\\ x\\ (Sec^2\\ x)\\ +\\ 3\\ Sin\\ x\\ Tan\\ x\\  x^2\\  +\\ x^3\\ Tan\\ x\\ Cos\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{x\\ +\\ 6}{x\\ -\\ 7}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x\\ +\\ 6),\\ \\hspace{5cm}\\ v\\ =\\ (x\\ -\\ 7)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(x\\ -\\ 7)\\ \\frac{d}{dx}\\ (x\\ +\\ 6)\\  -\\ (x\\ +\\ 6)\\ \\frac{d}{dx}\\ (x\\ -\\ 7)}{(x\\ -\\ 7)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(x\\ -\\ 7)\\ (1\\ +\\ 0)\\  -\\ (x\\ +\\ 6)\\  (1\\ -\\ 0)}{(x\\ -\\ 7)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{x\\ -\\ 7\\   -\\ x\\ -\\ 6}{(x\\ -\\ 7)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{-\\ 13}{(x\\ -\\ 7)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/-xoxngde4nc\" title=\"Differentiation - Part - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\" \"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13.} \\color {red} {Find \\frac{dy}{dx}}\\ (i)\\ if\\  y\\ =\\  (x\\ +\\ 1)(x\\ +\\ 2)(x\\ +\\ 3)\\ \\hspace{2cm}\\  (ii)\\ y\\ =\\ \\frac{x^2\\ +\\ 1}{e^x}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ June\\ 2022,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\   (i)\\  y\\ =\\ (x\\ +\\ 1)(x\\ +\\ 2)(x\\ +\\ 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x\\ +\\ 1,\\ \\hspace{2cm}\\ v\\ =\\ x\\ +\\ 2\\ \\hspace{2cm}\\ w\\ =\\ x\\ +\\ 3\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  (x\\ +\\ 1)\\ (x\\ +\\ 2)\\ \\frac{d}{dx}\\ (x\\ +\\ 3 )\\ +\\ (x\\ +\\ 2)\\ (x\\ +\\ 3)\\ \\frac{d}{dx}\\ (x\\ +\\ 1)\\  +\\ (x\\ +\\ 3)\\ (x\\ +\\ 1)\\ \\frac{d}{dx}\\ (x\\ +\\ 3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  (x\\ +\\ 1)\\ (x\\ +\\ 2)\\ (1\\ +\\ 0)\\ +\\ (x\\ +\\ 2)\\ (x\\ +\\ 3)\\ (1\\ +\\ 0)\\  +\\ (x\\ +\\ 3)\\ (x\\ +\\ 1)\\ (1\\ +\\ 0)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  (x\\ +\\ 1)\\ (x\\ +\\ 2)\\  +\\ (x\\ +\\ 2)\\ (x\\ +\\ 3)\\  +\\ (x\\ +\\ 3)\\ (x\\ +\\ 1)\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{x^2\\ +\\ 1}{e^x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x^2\\ +\\ 1),\\ \\hspace{5cm}\\ v\\ =\\ (e^x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{e^x\\ \\frac{d}{dx}\\ (x^2\\ +\\ 1)\\  -\\ (x^2\\ +\\ 1)\\ \\frac{d}{dx}\\ (e^x)}{(e^x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{e^x\\ (2\\ x)\\ (x^2\\ +\\ 1)\\  -\\ (x^2\\ +\\ 1)\\ e^x}{(e^x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/u5-a6l-Rb2Y\" title=\"Differentiation - Part - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\" \"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14.:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ (i)\\ if\\  y\\ =\\ e^x\\ log\\ x\\ Cos\\ x\\ (ii)\\ y\\ =\\ \\frac{x^2\\ +\\ Tan\\ x}{x\\ -\\ Sin\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\   (i)\\  y\\ =\\ e^x\\ log\\ x\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{2cm}\\ v\\ =\\ log\\ x\\ \\hspace{2cm}\\ w\\ =\\ Cos\\ x\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  e^x\\ log\\ x\\ \\frac{d}{dx}\\ (Cos\\ x)\\ +\\ log\\ x\\ Cos\\ x\\ \\frac{d}{dx}\\ (e^x)\\  +\\ Cos\\ x\\ e^x\\ \\frac{d}{dx}\\ (log\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  e^x\\ log\\ x\\ (-\\ Sin\\ x)\\ +\\ log\\ x\\ Cos\\ x\\ e^x\\  +\\ Cos\\ x\\ e^x\\ \\frac{1}{x}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  -\\ e^x\\ log\\ x\\ Sin\\ x\\ +\\ log\\ x\\ Cos\\ x\\ e^x\\  +\\ Cos\\ x\\ e^x\\ \\frac{1}{x}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{x^2\\ +\\ Tan\\ x}{x\\ -\\ Sin\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x^2\\ +\\ Tan\\ x),\\ \\hspace{5cm}\\ v\\ =\\ (x\\ -\\ Sin\\ x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(x\\ -\\ Sin\\ x)\\ \\frac{d}{dx}\\ (x^2\\ +\\ Tan\\ x)\\  -\\ (x^2\\ +\\ Tan\\ x)\\ \\frac{d}{dx}\\ (x\\ -\\ Sin\\ x)}{(x\\ -\\ Sin\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(x\\ -\\ Sin\\ x)\\ (2\\ x\\ +\\ Sec^2\\ x)\\  -\\ (x^2\\ +\\ Tan\\ x)\\  (1\\ -\\ Cos\\ x)}{(x\\ -\\ Sin\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ (i)\\ if\\  y\\ =\\ e^x\\ log\\ x\\ \\sqrt{x}\\ (ii)\\ y\\ =\\ \\frac{sin\\ x}{1\\ +\\ cos\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\   (i)\\  y\\ =\\ e^x\\ log\\ x\\ \\sqrt{x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{2cm}\\ v\\ =\\ log\\ x\\ \\hspace{2cm}\\ w\\ =\\ \\sqrt{x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  e^x\\ log\\ x\\ \\frac{d}{dx}\\ (\\sqrt{x})\\ +\\ log\\ x\\ \\sqrt{x}\\ \\frac{d}{dx}\\ (e^x)\\  +\\ \\sqrt{x}\\ e^x\\ \\frac{d}{dx}\\ (log\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  e^x\\ log\\ x\\ (\\frac{1}{2\\sqrt{x}})\\ +\\ log\\ x\\ \\sqrt{x}\\ e^x\\  +\\ \\sqrt{x}\\ e^x\\ \\frac{1}{x}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/QwLhJVNYnSI?list=PLQIom4Rz29vx-SPSIOXKbkeXEhzCwLBWq\" title=\"Differentiation - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{sin\\ x}{1\\ +\\ cos\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (Sin\\ x),\\ \\hspace{5cm}\\ v\\ =\\ (1\\ +\\ Cos\\ x)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(1\\ -\\ Cos\\ x)\\ \\frac{d}{dx}\\ (Sin\\ x)\\  -\\ (Sin\\ x)\\ \\frac{d}{dx}\\ (1\\ +\\ Cos\\ x)}{(1\\ +\\ Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(1\\ -\\ Cos\\ x)\\ (Cos\\ x)\\  -\\ (Sin\\ x)\\  (-\\ Sin\\ x)}{(1\\ +\\ Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{Cos\\ x\\  +\\ cos^2\\ x\\ +\\ sin^2\\ x}{(1\\ +\\ Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{Cos\\ x\\  +\\ 1}{(1\\ +\\ Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{1}{(1\\ +\\ Cos\\ x)}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/ENMIHhszD0U\" title=\"Differentiation - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16.}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ (i)\\ if\\  y\\ =\\ log\\ x\\ (2x\\ +\\ 1)\\ \\sqrt{x} \\ \\hspace{2cm}\\ (ii)\\ y\\ =\\ \\frac{ax\\ +\\ b}{cx\\ +\\ d}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\   (i)\\  y\\ =\\ log\\ x\\ (2x\\ +\\ 1)\\ \\sqrt{x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ log\\ x,\\ \\hspace{2cm}\\ v\\ =\\ 2x\\ +\\ 1\\ \\hspace{2cm}\\ w\\ =\\ \\sqrt{x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (u\\ v\\ w)\\ =\\  u\\ v\\ \\frac{dw}{dx}\\ +\\ v\\ w\\ \\frac{du}{dx}\\ +\\ w\\ u\\ \\frac{dv}{dx}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  log\\ x\\ (2x\\ +\\ 1)\\ \\frac{d}{dx}\\ (\\sqrt{x})\\ +\\ (2x\\ +\\ 1)\\ \\sqrt{x}\\ \\frac{d}{dx}\\ (log\\ x)\\  +\\ \\sqrt{x}\\  log\\ x\\ \\frac{d}{dx}\\ (2x\\ +\\ 1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  log\\ x\\ (2x\\ +\\ 1)\\  log\\ x\\ (2x\\ +\\ 1)\\  +\\ (2x\\ +\\ 1)\\ \\frac{1}{\\sqrt{x}}\\  +\\ 2\\sqrt{x}\\  log\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{ax\\ +\\ b}{cx\\ +\\ d}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (ax\\ +\\ b),\\ \\hspace{5cm}\\ v\\ =\\ (cx\\ +\\ d)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[W.\\ K.\\ T\\  \\frac{d}{dx}\\ (\\frac{u}{v})\\ =\\  \\frac{v\\ \\frac{du}{dx}\\ -\\ u\\ \\frac{dv}{dx}}{v^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(cx\\ +\\ d)\\ \\frac{d}{dx}\\ (ax\\ +\\ b)\\  -\\ (ax\\ +\\ b)\\ \\frac{d}{dx}\\ (cx\\ +\\ d)}{(cx\\ +\\ d)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(cx\\ +\\ d)\\ (a\\ +\\ 0)\\  -\\ (ax\\ +\\ b)\\  (c\\ +\\ 0)}{(cx\\ +\\ d)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{ad\\  -\\ bc}{(cx\\ +\\ d)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<h2 class=\"wp-block-heading has-text-align-center\">Exercise Problems<\/h2>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {1.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\  y\\ =\\  7\\ e^x\\ \\ +\\ 4\\ log\\ x\\ +\\ \\frac{1}{x^2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {2.} \\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\  y\\ =\\  8\\ e^x\\ -\\ 4\\ cosec\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {3.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\  y\\ =\\ e^x\\ Cos\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {4.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\  y\\ =\\ x^4\\ Sin\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {5.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ if\\  y\\ =\\ x^3\\ log\\ x\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {6.}\\ If\\  y\\ =\\ (x\\ +\\ 3)\\ (x\\ -\\ 4)\\  \\color {red} {find\\ \\frac{dy}{dx}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {7.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\  if\\  y\\ =\\ x^2\\  e^x\\ Sin\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {8.}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\  if\\  y\\ =\\ (x^2\\ +\\ 5)\\ Cos\\ x\\ e^{-\\ 2x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {9.}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\  if\\  y\\ =\\ \\frac{Sin\\ x}{log\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {10.}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\  if\\  y\\ =\\ \\frac{x\\ Sin\\ x}{e^x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {11.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ (i)\\ if\\  y\\ =\\ x\\ e^x\\ log\\ x\\ (ii)\\ y\\ =\\ \\ (x^2\\ +\\ 2)\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {12.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ (i)\\ if\\  y\\ =\\ e^x\\ x\\ Cos\\ x\\ (ii)\\ y\\ =\\ \\frac{x\\ +\\ Sin\\ x}{1\\ -\\ Cos\\ x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {13.}\\ \\color {red} {Find}\\ \\frac{dy}{dx}\\ (i)\\ if\\  y\\ =\\ (2\\ x\\ +\\ 1)(3\\ x\\ -\\ 7)(4\\ -\\ 9\\ x)\\ \\hspace{2cm}\\  (ii)\\ y\\ =\\ \\frac{e^x\\ +\\ Sin\\ x}{1\\ -\\ Cos\\ x}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {14:}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\ (i)\\ if\\  y\\ =\\ log\\ x\\ (2x\\ +\\ 1)\\ \\sqrt{x} \\ \\hspace{2cm}\\ (ii)\\ y\\ =\\ \\frac{ax\\ +\\ b}{cx\\ +\\ d}\\ \\hspace{15cm}\\] <\/div>\n","protected":false},"excerpt":{"rendered":"<p>Exercise 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