{"id":24734,"date":"2022-01-03T16:32:12","date_gmt":"2022-01-03T11:02:12","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=24734"},"modified":"2024-07-08T18:52:29","modified_gmt":"2024-07-08T13:22:29","slug":"differentiation-excercise","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=24734","title":{"rendered":"DIFFERENTIATION (Excercise)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<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 {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 {blue}{Solution:}\\  y\\ =\\ 7\\ e^x\\ \\ +\\ 4\\ log\\ x\\ +\\ \\frac{1}{x^2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 7\\ \\frac{d}{dx}(e^x)\\ +\\ 4\\ \\frac{d}{dx}(log\\ x)\\ +\\ \\frac{d}{dx}(x^{-2})\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 7\\  e^x\\ +\\ 4\\ \\frac{1}{x}\\ +\\ (-\\ 2)\\ x^{-2\\ -\\ 1}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 7\\  e^x\\ +\\ \\frac{4}{x}\\ -\\ 2\\  x^{-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\"><\/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 {blue}{Solution:}\\  y\\ =\\ 8\\ e^x\\ -\\ 4\\ cosec\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ 8\\ \\frac{d}{dx}(e^x)\\ +\\ \\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}\\ =\\ 8\\  e^x\\ +\\ 3\\ x^2\\ -\\ Sin\\ x\\ \\hspace{10cm}\\]<\/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 {blue}{Solution:}\\  y\\ =\\ e^x\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{5cm}\\ v\\ =\\ Cos\\ 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}(Cos\\ x)\\ +\\ Cos\\ x\\ \\frac{d}{dx}(e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ e^x\\ (-\\ sin\\ x)\\ +\\ Cos\\ x\\ e^x\\ \\hspace{10cm}\\]<\/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<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ x^4\\ Sin\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^4,\\ \\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}\\ =\\ x^4\\ \\frac{d}{dx}(Sin\\ x)\\ +\\ Sin\\ x\\ \\frac{d}{dx}(x^4)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^4\\ Cos\\ x\\ +\\ Sin\\ x\\ 4(x^3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x^4\\ Cos\\ x\\ +\\ 4\\ Sin\\ x\\ x^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\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/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 {blue}{Solution:}\\     y\\ =\\ x^3\\  log\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^3,\\ \\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}\\ =\\ x^3\\ \\frac{d}{dx}(log\\ x)\\ +\\  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\\ (\\frac{1}{x})\\ +\\ 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\\ log\\ x)\\hspace{10cm}\\]<\/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 {blue}{Solution:}\\  y\\ =\\ (x\\ +\\ 3)\\ (x\\ -\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x\\ +\\ 3),\\ \\hspace{5cm}\\ v\\ =\\ (x\\ -\\ 4)\\]<\/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}(x\\ -\\ 4)\\ +\\ (x\\ -\\ 4)\\ \\frac{d}{dx}(x\\ +\\ 3)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ (x\\ +\\ 3)\\ (1)\\ +\\ (x\\ -\\ 4)\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ x\\ +\\ 3\\ +\\ x\\ -\\ 4\\ \\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} {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 {blue}{Solution:}\\   y\\ =\\ x^2\\  e^x\\ Sin\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^2,\\ \\hspace{2cm}\\ v\\ =\\ e^x\\ \\hspace{2cm}\\ w\\ =\\ Sin\\ 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^2\\ e^x\\ \\frac{d}{dx}\\ (Sin\\ x)\\ +\\  e^x\\ Sin\\ x\\ \\frac{d}{dx}\\ (x^2)\\  +\\ Sin\\ x\\ x^2\\ \\frac{d}{dx}\\ (e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   x^2\\ e^x\\  Cos\\ x\\ +\\ e^x\\ Sin\\ x\\ (2\\ x) +\\ Sin\\ x\\ x^2\\ e^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\">\\[\\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 {blue}{Solution:}\\   y\\ =\\ (x^2\\ +\\ 5)\\ Cos\\ x\\ e^{-\\ 2x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x^2\\ +\\ 5,\\ \\hspace{2cm}\\ v\\ =\\ Cos\\ x\\ \\hspace{2cm}\\ w\\ =\\ e^{-\\ 2x}\\]<\/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^2\\ +\\ 5)\\ Cos\\ x\\ \\frac{d}{dx}\\ (e^{-\\ 2x})\\ +\\  Cos\\ x\\ e^{-\\ 2x}\\ \\frac{d}{dx}\\ (x^2\\ +\\ 5)\\  +\\ e^{-\\ 2x}\\ (x^2\\ +\\ 5)\\ \\frac{d}{dx}\\ (Cos\\ x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   (x^2\\ +\\ 5)\\ Cos\\ x\\ (-\\ 2\\ e^{-\\ 2x})\\ +\\ Cos\\ x\\ e^{-\\ 2x}\\ (2\\ x)\\ +\\ e^{-\\ 2x}\\ (x^2\\ +\\ 5)\\ (-\\ Sin\\ x)\\ \\hspace{10cm}\\]<\/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 {blue}{Solution:}\\  y\\ =\\ \\frac{Sin\\ x}{log\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ Sin\\ x,\\ \\hspace{5cm}\\ v\\ =\\ log\\ 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{log\\ x\\ \\frac{d}{dx}\\ (Sin\\ x)\\  -\\ (Sin\\ x)\\ \\frac{d}{dx}\\ (log\\ x)}{(log\\ x)^2}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\   \\frac{(log\\ x)\\ (Cos\\ x)\\  -\\ (Sin\\ x)\\  (\\frac{1}{x})}{(log\\ 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\">\\[\\color {purple} {10.}\\ \\color {red} {Find\\ \\frac{dy}{dx}}\\  if\\  y\\ =\\ \\frac{x\\ Sin\\ x}{e^x}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  y\\ =\\ \\frac{x\\ Sin\\ x}{e^x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x\\ Sin\\ x,\\ \\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\\ Sin\\ x)\\  -\\ (Sin\\ x)\\ \\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)\\ (x\\ Cos\\ x)\\  -\\ (Sin\\ x)\\  (e^x)}{(e^x)^2}\\ \\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<!-- 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 {blue}{Solution:}\\   (i)\\  y\\ =\\ x\\ e^x\\ log\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ x,\\ \\hspace{2cm}\\ v\\ =\\ e^x\\ \\hspace{2cm}\\ w\\ =\\ log\\ 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\\ e^x\\ \\frac{d}{dx}\\ (log\\ x)\\ +\\ e^x\\ log\\ x\\ \\frac{d}{dx}\\ (x)\\  +\\ log\\ x\\ x\\ \\frac{d}{dx}\\ (e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  x\\ e^x\\ (\\frac{1}{x})\\ +\\ e^x\\ log\\ x\\ (1)\\  +\\ log\\ x\\ x\\ (e^x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  \\frac{x\\ e^x}{x}\\ +\\ e^x\\ log\\ x\\   +\\ x\\ log\\ x\\ e^x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ (x^2\\ +\\ 2)\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x^2\\ +\\ 2),\\ \\hspace{5cm}\\ v\\ =\\ Cos\\ 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^2\\ +\\ 2)\\ \\frac{d}{dx}(Cos\\ x)\\ +\\ Cos\\ x\\ \\frac{d}{dx}(x^2\\ +\\ 2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ (x^2\\ +\\ 2)\\ (-\\ Sin\\ x)\\ +\\ Cos\\ x\\ (2\\ x\\ +\\ 0)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\frac{dy}{dx}\\ =\\ -(x^2\\ +2)\\ Sin\\ x\\ +\\ 2\\ x\\  Cos\\ x\\hspace{10cm}\\]<\/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 {blue}{Solution:}\\   (i)\\  y\\ =\\ e^x\\ x\\ Cos\\ x\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ e^x,\\ \\hspace{2cm}\\ v\\ =\\ 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\\ x\\ \\frac{d}{dx}\\ (Cos\\ x)\\ +\\ x\\ Cos\\ x\\ \\frac{d}{dx}\\ (e^x)\\  +\\ Cos\\ x\\ e^x\\ \\frac{d}{dx}\\ (x)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  e^x\\ x\\ (-\\ Sin\\ x)\\ +\\ x\\ Cos\\ x\\ e^x\\  +\\ Cos\\ x\\ e^x\\ (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\frac{dy}{dx}\\ =\\  -\\ e^x\\ x\\ Sin\\ x\\ +\\ x\\ Cos\\ x\\ e^x\\  +\\ Cos\\ x\\ e^x\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(ii)\\  y\\ =\\ \\frac{x\\ +\\ Sin\\ x}{1\\ -\\ Cos\\ x}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ (x\\ +\\ 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}\\ (x\\ +\\ Sin\\ x)\\  -\\ (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)\\ (1\\ +\\ Cos\\ x)\\  -\\ (x\\ +\\ Sin\\ x)\\  (Sin\\ x)}{(1\\ -\\ Cos\\ x)^2}\\ \\hspace{10cm}\\]<\/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 {blue}{Solution:}\\   (i)\\  y\\ =\\ (2\\ x\\ +\\ 1)(3\\ x\\ -\\ 7)(4\\ -\\ 9\\ x)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Here\\ u\\ =\\ 2\\ x\\ +\\ 1,\\ 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