{"id":23689,"date":"2021-12-03T21:22:41","date_gmt":"2021-12-03T15:52:41","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=23689"},"modified":"2024-07-13T12:15:11","modified_gmt":"2024-07-13T06:45:11","slug":"partial-differentiation","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=23689","title":{"rendered":"PARTIAL DIFFERENTIATION (Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition}:\\ \\hspace{20cm}\\]<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\">\\[Let\\ u\\ =\\ f ( x , y )\\  then\\ the\\ partial\\ differentiation\\ of\\ u\\ with\\ respect\\ to\\  x\\  is\\ defined\\ as\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[differentiation\\ of\\ u\\ w.\\ r.\\ t\\  x\\  treating\\  y\\  as\\ constant\\ and\\ is\\ denoted\\ by\\ \\frac{\u2202u}{\u2202x}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Similarly\\  partial\\ differentiation\\ of\\ u\\ with\\ respect\\ to\\  y\\  is\\ defined\\ as\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[differentiation\\ of\\ u\\ w.\\ r.\\ t\\  y\\  treating\\  x\\  as\\ constant\\ and\\ is\\ denoted\\ by\\ \\frac{\u2202u}{\u2202y}\\]<\/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:inline-block;width:728px;height:90px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Problems}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\ \\color {red} {Find\\ \\frac{\u2202u}{\u2202x}\\ and\\ \\frac{\u2202u}{\u2202y}}\\ if\\ u\\ = x^2\\ +\\ y^2\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ = x^2\\ +\\ y^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ \\frac{\u2202}{\u2202x}( y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 2\\ x\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 2\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = x^2\\ +\\ y^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ \\frac{\u2202}{\u2202y}( x^2)\\ +\\ \\frac{\u2202}{\u2202y}( y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 2\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 2\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ If\\ u\\ = x^3\\ +\\ y^3\\ +\\ 3\\ x^2\\ y\\ +\\ 3\\ x\\ y^2,\\ \\color {red} {find\\ \\frac{\u2202u}{\u2202x}\\ and\\ \\frac{\u2202u}{\u2202y}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ = x^3\\ +\\ y^3\\ +\\ 3\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ \\frac{\u2202}{\u2202x}( y^3)\\ +\\ 3\\ \\frac{\u2202}{\u2202x}( x^2)\\ y\\ +\\ 3\\ y^2\\ \\frac{\u2202}{\u2202x}(x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ 0\\ +\\ 3\\ (2\\ x)\\ y)\\ +\\ 3\\ y^2\\ (1)\\ \\frac{\u2202}{\u2202x}( y))\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\  +\\ 6\\ x\\ y\\ +\\ 3\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = x^3\\ +\\ y^3\\ +\\ 3\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ \\frac{\u2202}{\u2202y}( x^3)\\ +\\ \\frac{\u2202}{\u2202y}( y^3)\\ +\\ 3\\ x^2\\ \\frac{\u2202}{\u2202y}(y)\\ +\\ 3\\ x\\ \\frac{\u2202}{\u2202y}(y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 3\\ y^2\\ +\\ 3\\ x^2\\ (1)\\ +\\ 3\\ x\\ (2\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 3\\ y^2\\ +\\ 3\\ x^2\\ +\\ 6\\ x\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ If\\ u\\ = 2\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 2\\ x\\ y\\ ,\\ \\color {red} {find\\ \\frac{\u2202u}{\u2202x}\\ and\\ \\frac{\u2202u}{\u2202y}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ = 2\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 2\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ 4\\ \\frac{\u2202}{\u2202x}( y^3)\\ +\\ 2\\ y\\ \\frac{\u2202}{\u2202x}( x)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 2\\ (3\\ x^2)\\ +\\ 0\\ +\\ 2\\ y\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 6\\ x^2\\ +\\ 2\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = 2\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 2\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202y}( x^3)\\ +\\ 4\\ \\frac{\u2202}{\u2202y}( y^3)\\ +\\ 2\\ x\\ \\frac{\u2202}{\u2202y}( y)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 2\\ (0)\\ +\\ 4(3\\ y^2)\\ +\\ 2\\ x\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 12\\ y^2\\ +\\ 2\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 12\\ y^2\\ +\\ 2\\ x\\ \\hspace{10cm}\\]\n<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/majkzlYb7G4\" title=\"Partial 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\\ 4.}\\ If\\ u\\ = 3\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 6\\ x\\ y\\ ,\\ \\color {red} {find\\ (i)\\ \\frac{\u2202u}{\u2202x}\\ (ii)\\ \\frac{\u2202u}{\u2202y}}\\ \\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:}\\  (i)\\ Given\\ u\\ = 3\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 6\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ 3\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ 4\\ \\frac{\u2202}{\u2202x}( y^3)\\ +\\ 6\\ y\\ \\frac{\u2202}{\u2202x}( x)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ (3\\ x^2)\\ +\\ 0\\ +\\ 6\\ y\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 9\\ x^2\\ +\\ 6\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\ u\\ = 3\\ x^3\\ +\\ 4\\ \\ y^3\\ +\\ 6\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ 3\\ \\frac{\u2202}{\u2202y}( x^3)\\ +\\ 4\\ \\frac{\u2202}{\u2202y}( y^3)\\ +\\ 6\\ x\\ \\frac{\u2202}{\u2202y}( y)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 3\\ (0)\\ +\\ 4(3\\ y^2)\\  +\\ 6\\ x\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 12\\ y^2\\ +\\ 6\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 12\\ y^2\\ +\\ 6\\ 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<!-- 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\\ 5.}\\ If\\ u\\ = 2\\ x^3\\ +\\ 3\\ x^2y\\ +\\ 4\\ x y^2\\ +\\ 4\\ y^3,\\ \\color {red} {find\\ \\frac{\u2202u}{\u2202x}\\ and\\ \\frac{\u2202u}{\u2202y}}\\ \\hspace{15cm}\\] <\/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 {blue}{Solution:}\\  Given\\ u\\ = 2\\ x^3\\ +\\ 3\\ x^2y\\ +\\ 4\\ x y^2\\ +\\ 4\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ 3\\ y\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ 4\\ y^2\\ \\frac{\u2202}{\u2202x}( x)\\ +\\ 4\\ \\frac{\u2202}{\u2202x}(y^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 2\\ (3\\ x^2)\\ +\\ 3y\\ (2x)\\ +\\ 4\\ y^2\\ (1)\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 6\\ x^2\\ +\\ 6\\ xy\\ +\\ 4\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = 2\\ x^3\\ +\\ 3\\ x^2y\\ +\\ 4\\ x y^2\\ +\\ 4\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202y}( x^3)\\ +\\ 3\\ x^2\\ \\frac{\u2202}{\u2202y}(y)\\ +\\ 4\\ x\\ \\frac{\u2202}{\u2202y}(y^2)\\ +\\ 4\\ \\frac{\u2202}{\u2202y}(y^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 2\\ (0)\\ +\\ 3\\ x^2\\ (1)\\ +\\ 4\\ x\\ (2y)\\ +\\ 4(3y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 3\\ x^2\\ +\\ 8\\ xy\\ +\\ 12\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\  3\\ x^2\\ +\\ 8\\ xy\\ +\\ 12\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/9HRHqgF-05k\" title=\"Partial Differentiation - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"true\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ If\\ u\\ =\\ x^3\\  +\\ y^3\\ +\\ xy ,\\ find\\ \\frac{\u2202^2u}{\u2202x^2}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ =\\  x^3\\  +\\ y^3\\ +\\ xy\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ \\frac{\u2202}{\u2202x}(y^3)\\ +\\ \\frac{\u2202}{\u2202x}(xy)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ 0\\ +\\ y \\frac{\u2202}{\u2202x}(x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}(\\frac{\u2202u}{\u2202x})\\ =\\ \\frac{\u2202}{\u2202x}(3\\ x^2\\ +\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\  \\frac{\u2202}{\u2202x}(3\\ x^2)\\ +\\ \\frac{\u2202}{\u2202x}(y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ (2\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 6\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 7:}\\ If\\ u\\ =\\ log(x^2\\ +\\ y^2)\\ ,\\ \\color {red} {find\\ \\frac{\u2202u}{\u2202x}\\ ,\\ \\frac{\u2202u}{\u2202y}\\ and\\ \\frac{\u2202^2u}{\u2202y\\ \u2202x}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ =\\ log(x^2\\ +\\ y^2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}(log(x^2\\ +\\ y^2))\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{1}{x^2\\ +\\ y^2}\\ \\frac{\u2202}{\u2202x}(x^2\\ +\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{1}{x^2\\ +\\ y^2}\\ (2\\ x\\ +\\ 0)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{2\\ x}{x^2\\ +\\ y^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ =\\ log(x^2\\ +\\ y^2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ \\frac{\u2202}{\u2202y}(log(x^2\\ +\\ y^2))\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{1}{x^2\\ +\\ y^2}\\ \\frac{\u2202}{\u2202y}(x^2\\ +\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{1}{x^2\\ +\\ y^2}\\ (0\\ +\\ 2\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{2\\ y}{x^2\\ +\\ y^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2u}{\u2202y\\ \u2202x}\\ =\\ \\frac{\u2202}{\u2202y}(\\frac{2\\ x}{x^2\\ +\\ y^2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2u}{\u2202y\\ \u2202x}\\ =\\ \\frac{(x^2\\ +\\ y^2)\\ \\frac{\u2202}{\u2202y}(2\\ x)\\ -\\ 2\\ x\\ \\frac{\u2202}{\u2202y}(x^2\\ +\\ y^2)}{(x^2\\ +\\ y^2)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2u}{\u2202y\\ \u2202x}\\ =\\ \\frac{(x^2\\ +\\ y^2)\\ (0)\\ -\\ 2\\ x\\ (0\\ +\\ 2\\ y)}{(x^2\\ +\\ y^2)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2u}{\u2202y\\ \u2202x}\\ =\\ \\frac{-\\ 4\\ x\\ y}{(x^2\\ +\\ y^2)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\ If\\ u\\ =\\ 2\\ x^3\\ -\\ 3\\ x^2y\\ +\\ 3\\ x y^2\\ +\\ 5\\ y^3,\\  \\color {red} {find\\ the\\ value\\ of\\ \\ x\\ \\frac{\u2202u}{\u2202x}\\ +\\ y\\ \\frac{\u2202u}{\u2202y}}\\  \\hspace{15cm}\\] <\/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 {blue}{Solution:}\\  Given\\ u\\ = 2\\ x^3\\ -\\ 3\\ x^2y\\ +\\ 3\\ x y^2\\ +\\ 5\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202x}( x^3)\\ -\\ 3\\ y\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ 3\\ y^2\\ \\frac{\u2202}{\u2202x}( x)\\ +\\ 5\\ \\frac{\u2202}{\u2202x}(y^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 2\\ (3\\ x^2)\\ -\\ 3y\\ (2x)\\ +\\ 3\\ y^2\\ (1)\\ +\\ 5(0)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 6\\ x^2\\ -\\ 6\\ xy\\ +\\ 3\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = 2\\ x^3\\ -\\ 3\\ x^2y\\ +\\ 3\\ x y^2\\ +\\ 5\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ 2\\ \\frac{\u2202}{\u2202y}( x^3)\\ -\\ 3\\ x^2\\ \\frac{\u2202}{\u2202y}(y)\\ +\\ 3\\ x\\ \\frac{\u2202}{\u2202y}(y^2)\\ +\\ 5\\ \\frac{\u2202}{\u2202y}(y^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 2\\ (0)\\ -\\ 3\\ x^2\\ (1)\\ +\\ 3\\ x\\ (2y)\\ +\\ 5(3y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ -\\ 3\\ x^2\\ +\\ 6\\ xy\\ +\\ 15\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ x (6\\ x^2\\ -\\ 6\\ xy\\ +\\ 3\\ y^2)\\ +\\ y(-\\ 3\\ x^2\\ +\\ 6\\ xy\\ +\\ 15\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 6\\ x^3\\ -\\ 6\\ x^2\\ y\\ +\\  3\\ x\\ y^2\\ -\\ 3\\ x^2\\ y\\ +\\ 6\\ x\\ y^2\\ +\\ 15\\ y^3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 6\\ x^3\\ -\\ 9\\ x^2\\ y\\ +\\ 9\\ x\\ y^2\\ +\\ 15\\ y^3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 6\\ x^3\\ -\\ 9\\ x^2\\ y\\ +\\ 9\\ x\\ y^2\\ +\\ 15\\ y^3\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/YxYRrlnz_3g\" title=\"Partial 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\\ 9:}\\ If\\ u\\ =\\ x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2\\  ,\\ \\color {red} {prove\\ that\\ x\\ \\frac{\u2202u}{\u2202x}\\ +\\ y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 3\\ u}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ =\\  x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2\\ &#8212;&#8212;&#8211;\\ (1) \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  &#8216;\\ x\\ &#8216;\\ \\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{\u2202}{\u2202x}(x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ \\frac{\u2202}{\u2202x}( y^3)\\ +\\ 3\\ y^2\\ \\frac{\u2202}{\u2202x}( x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ 0\\ + 3\\ y^2\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ + 3\\ y^2\\ &#8212;&#8212;\\ (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ =\\ x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  &#8216;\\ y\\ &#8216;\\ \\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{\u2202}{\u2202y}(x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{\u2202}{\u2202y}( x^3)\\ +\\ \\frac{\u2202}{\u2202y}( y^3)\\ +\\ 3\\ x\\ \\frac{\u2202}{\u2202y}(y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 3\\ y^2\\ + 3\\ x\\ (2y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 3\\ y^2\\ + 6\\ x\\ y\\ &#8212;&#8212;\\ (3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ x (3\\ x^2\\ + 3\\ y^2)\\ +\\ y(3\\ y^2\\ + 6\\ x\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3\\ x^3\\ +\\ 3\\ x\\ y^2\\ +\\ 3\\ y^3\\ +\\ 6\\ x\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3\\ x^3\\ +\\ 3\\ y^3\\ +\\ 9\\ x\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3(x^3\\ +\\ y^3\\ +\\ 3\\ x\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3\\ (u)\\ &#8212;&#8211; using (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 3\\ u\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\ If\\ u\\ = \\frac{x^3\\ y^3}{x^3\\ +\\ y^3} ,\\ \\color {red} {Show\\ that\\ x\\ \\frac{\u2202u}{\u2202x}\\ +\\ y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 3\\ u}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\  Given\\ u\\ =\\ \\frac{x^3\\ y^3}{x^3\\ +\\ y^3}\\ &#8212;&#8212;&#8211;\\ (1) \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  &#8216;\\ x\\ &#8216;\\ \\ on\\ both\\ sides\\ \\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\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{\u2202}{\u2202x}(\\frac{x^3\\ y^3}{x^3\\ +\\ y^3})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{(x^3\\ +\\ y^3)\\ \\frac{\u2202}{\u2202x}(x^3\\ y^3)\\ -\\ x^3\\ y^3\\ \\frac{\u2202}{\u2202x}(x^3\\ +\\ y^3)}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{(x^3\\ +\\ y^3)\\ (y^3\\ 3\\ x^2)\\ -\\ x^3\\ y^3\\ (3\\ x^2\\ +\\ 0)}{(x^3\\ +\\ y^3)^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\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{3\\ x^5\\ y^3\\ +\\ 3\\ x^2\\ y^6\\  -\\ 3\\ x^5\\ y^3}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ \\frac{3\\ x^2\\ y^6}{(x^3\\ +\\ y^3)^2}\\ &#8212;&#8212;\\ (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ =\\ \\frac{x^3\\ y^3}{x^3\\ +\\ y^3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  &#8216;\\ y\\ &#8216;\\ \\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{\u2202}{\u2202y}(\\frac{x^3\\ y^3}{x^3\\ +\\ y^3})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{(x^3\\ +\\ y^3)\\ \\frac{\u2202}{\u2202y}(x^3\\ y^3)\\ -\\ x^3\\ y^3\\ \\frac{\u2202}{\u2202y}(x^3\\ +\\ y^3)}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{(x^3\\ +\\ y^3)\\ (x^3\\ 3\\ y^2)\\ -\\ x^3\\ y^3\\ (0\\ +\\ 3\\ y^2)}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{3\\ x^6\\ y^2\\ +\\ 3\\ x^3\\ y^5\\  -\\ 3\\ x^3\\ y^5}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ \\frac{3\\ x^6\\ y^2}{(x^3\\ +\\ y^3)^2}\\ &#8212;&#8212;\\ (3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ x (\\frac{3\\ x^2\\ y^6}{(x^3\\ +\\ y^3)^2})\\ +\\ y(\\frac{3\\ x^6\\ y^2}{(x^3\\ +\\ y^3)^2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{3\\ x^3\\ y^6}{(x^3\\ +\\ y^3)^2}\\ +\\ \\frac{3\\ x^6\\ y^3}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{3\\ x^3\\ y^6\\ +\\ 3\\ x^6\\ y^3}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{3\\ x^3\\ y^3(y^3\\ +\\ x^3)}{(x^3\\ +\\ y^3)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ \\frac{3\\ x^3\\ y^3}{x^3\\ +\\ y^3}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3\\ (u)\\ &#8212;&#8211; using (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ \\frac{\u2202u}{\u2202x}\\ +y\\ \\frac{\u2202u}{\u2202y}\\ =\\ 3\\ u\\ \\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\\ 11:}\\ If\\ u\\ =\\ x^3\\ -\\ 2\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ +\\ y^3\\ ,\\ \\color {red} {find\\ \\frac{\u2202^2u}{\u2202x^2}\\ and\\ \\frac{\u2202^2u}{\u2202y^2}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ = x^3\\ -\\ 2\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ +\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}( x^3)\\ -\\ 2\\ y\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ 3\\ y^2\\ \\frac{\u2202}{\u2202x}( x)\\ +\\ \\frac{\u2202}{\u2202x}(y^3)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ -\\ 2\\ y\\ (2\\ x)\\ +\\ 3\\ y^2\\ (1)\\ +\\ 0\\ \\frac{\u2202}{\u2202x}( y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ -\\ 4\\ x\\ y\\ +\\ 3\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}(\\frac{\u2202u}{\u2202x})\\ =\\ \\frac{\u2202}{\u2202x}(3\\ x^2\\ -\\ 4\\ x\\ y\\ +\\ 3\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ \\frac{\u2202}{\u2202x}( x^2)\\ -\\ 4\\ y\\ \\frac{\u2202}{\u2202x}( x)\\ +\\ \\frac{\u2202}{\u2202x}(3\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ (2\\ x)\\ -\\ 4\\ y\\ (1)\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 6\\ x\\ -\\ 4\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ = x^3\\ -\\ 2\\ x^2\\ y\\ +\\ 3\\ x\\ y^2\\ +\\ y^3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}\\ (u)\\ =\\ \\frac{\u2202}{\u2202y}( x^3)\\ -\\ 2\\ x^2\\ \\frac{\u2202}{\u2202y}( y)\\ +\\ 3\\ x\\ \\frac{\u2202}{\u2202y}(y^2)\\ +\\ \\frac{\u2202}{\u2202y}(y^3)\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ -\\ 2\\ x^2\\ (1)\\ +\\ 3\\ x\\ (2\\ y)\\ +\\ 3\\ y^2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ -\\ 2\\ x^2\\ +\\ 6\\ x\\ y\\ +\\ 3\\ y^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\">\\[Again\\ Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}(\\frac{\u2202u}{\u2202y})\\ =\\ \\frac{\u2202}{\u2202y}(-\\ 2\\ x^2\\ +\\ 6\\ x\\ y\\ +\\ 3\\ y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ \\frac{\u2202}{\u2202y}( -2\\ x^2)\\ +\\ 6\\ x\\ \\frac{\u2202}{\u2202y}(y)\\ +\\ 3\\ \\frac{\u2202}{\u2202y}(y^2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ 0\\ +\\ 6\\ x\\ (1)\\ +\\ 3(2\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ 6\\ x\\ +\\ 6\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ If\\ u\\ =\\ x^3\\ +\\ y^3\\ +\\ 4\\ x\\ y,\\ \\color {red} {find\\ the\\ x^2\\ \\frac{\u2202^2u}{\u2202x^2}\\ +\\ y^2\\ \\frac{\u2202^2u}{\u2202y^2}}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ u\\ = x^3\\ +\\ y^3\\ +\\ 4\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}\\ (u)\\ =\\ \\frac{\u2202}{\u2202x}( x^3)\\ +\\ \\frac{\u2202}{\u2202x}(y^3)\\ +\\ 4\\ y\\ \\frac{\u2202}{\u2202x}( x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ 0\\ +\\ 4\\ y (1)\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202x}\\ =\\ 3\\ x^2\\ +\\ 4\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Again\\ Differentiate\\ partially\\ w.\\ r.\\ t.\\  x\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202x}(\\frac{\u2202u}{\u2202x})\\ =\\ \\frac{\u2202}{\u2202x}(3\\ x^2\\ +\\ 4\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ \\frac{\u2202}{\u2202x}( x^2)\\ +\\ 4 \\frac{\u2202}{\u2202x}(4\\ y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 3\\ (2\\ x)\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202x^2}\\ =\\ 6\\ x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[u\\ =\\  x^3\\ +\\ y^3\\ +\\ 4\\ x\\ y\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\ 0\\ +\\ 3\\ y^2\\ +\\ 4\\ x( 1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202u}{\u2202y}\\ =\\  3\\ y^2\\ +\\ 4\\ x\\ \\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\">\\[Again\\ Differentiate\\ partially\\ w.\\ r.\\ t.\\  y\\ on\\ both\\ sides\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202}{\u2202y}(\\frac{\u2202u}{\u2202y})\\ =\\ \\frac{\u2202}{\u2202y}(3\\ y^2\\ +\\ 4\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ 3\\ \\frac{\u2202}{\u2202y}( y^2)\\ +\\ \\frac{\u2202}{\u2202y}( 4\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ 3(2y)\\ +\\ 0\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\  6\\ y\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2\\ \\frac{\u2202^2\\ u}{\u2202x^2}\\ +y\\ \\frac{\u2202^2\\ u}{\u2202y^2}\\ =\\ x^2 (6\\ x)\\ +\\ y^2(y)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ x^3\\ +\\ 6\\ y^3\\ \\hspace{10cm}\\]\n\n<\/div>\n\n\n<p><iframe width=\"853\" height=\"480\" 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