{"id":4311,"date":"2024-01-30T12:20:48","date_gmt":"2024-01-30T06:20:48","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=4311"},"modified":"2024-11-05T16:26:12","modified_gmt":"2024-11-05T10:26:12","slug":"derivative-of-different-functions","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/derivative-of-different-functions\/","title":{"rendered":"\u09ad\u09bf\u09a8\u09cd\u09a8 \u09ad\u09bf\u09a8\u09cd\u09a8 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c (Determining the Derivative of different functions)"},"content":{"rendered":"<h2 class=\"\" data-sourcepos=\"1:1-1:18\"><span style=\"color: #339966;\">\u09ab\u09be\u0982\u09b6\u09a8 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7?<\/span><\/h2>\n<p data-sourcepos=\"3:1-3:21\"><strong>\u09ab\u09be\u0982\u09b6\u09a8<\/strong> \u09b9\u09b2 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09c7\u099f\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0995\u09c7 \u0985\u09a8\u09cd\u09af \u098f\u0995\u099f\u09bf \u09b8\u09c7\u099f\u09c7\u09b0 \u098f\u0995\u0995 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u0995\u09b0\u09c7 \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09af\u09bc\u09ae\u0964 \u09b8\u09b9\u099c \u0995\u09a5\u09be\u09df, \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09b2 \u098f\u0995 \u09a7\u09b0\u09a8\u09c7\u09b0 \u09af\u09a8\u09cd\u09a4\u09cd\u09b0 \u09af\u09be \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0987\u09a8\u09aa\u09c1\u099f \u09a6\u09bf\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0986\u0989\u099f\u09aa\u09c1\u099f \u09a6\u09c7\u09df\u0964<\/p>\n<h2><span style=\"color: #339966;\"><strong>\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c<\/strong><b>\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df (Determining the Derivative of Functions)<\/b><\/span><\/h2>\n<h3><span style=\"color: #800080;\"><b>Type-18: \u098f\u0995\u09be\u09a6\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09b8\u09ae\u09a8\u09cd\u09ac\u09df \u09b8\u09ae\u09be\u09a7\u09be\u09a8\u09c7 \u09b2\u0997\u09be\u09b0\u09bf\u09a6\u09ae <\/b><b>(Logarithm to solve the combination of one function)<\/b><\/span><\/h3>\n<p><strong>Concept:<\/strong><span style=\"font-weight: 400;\"> \u0985\u09a8\u09c7\u0995\u0997\u09c1\u09b2\u09bf<span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/academic\/10\/\"> \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0<\/a><\/span> \u0997\u09c1\u09a3 \u09ac\u09be \u09ad\u09be\u0997 \u0986\u0995\u09be\u09b0\u09c7 \u09ab\u09be\u0982\u09b6\u09be\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09b8\u09b9\u0997 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0989\u09ad\u09df \u09aa\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">ln<\/span> \u09a8\u09bf\u09a4\u09c7 \u09b9\u09ac\u09c7<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><b>Example-60:<\/b> <strong><span class=\"katex-eq\" data-katex-display=\"false\">(\\sin x),(\\ln x) \\cdot(\\tan x)\\left(e^{x}\\right)<\/span> \u098f\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u0995<\/strong><b> \u09b8\u09b9\u0997 \u09ac\u09c7\u09b0 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7<\/b><b>\u0964<\/b><\/p>\n<p><b><span class=\"katex-eq\" data-katex-display=\"false\"> \\mathrm{Sol}^{n}<\/span> : <\/b><span style=\"font-weight: 400;\">\u09a7\u09b0\u09bf, <span class=\"katex-eq\" data-katex-display=\"false\">y=(\\sin x)(\\ln x)(\\tan x)\\left(e^{x}\\right)<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{l}\n\n\\Rightarrow \\ln y=\\ln (\\sin x)+\\ln (\\ln x)+\\ln (\\tan x)+\\ln \\left(e^{x}\\right) \\\\\n\n\\therefore \\frac{1}{y} \\cdot \\frac{d y}{d x}=\\frac{1}{\\sin x} \\cdot \\cos x+\\frac{1}{\\ln x} \\cdot \\frac{1}{x}+\\frac{1}{\\tan x} \\cdot \\sec ^{2} x+1 \\\\\n\n\\Rightarrow \\frac{d y}{d x}=y\\left[\\cot x+\\frac{1}{x \\ln x}+\\frac{\\sec x}{\\sin x}+1\\right] \\\\\n\n\\therefore \\frac{d y}{d x}=(\\sin x)(\\ln x)(\\tan x)\\left(e^{x}\\right)\\left[\\cot x+\\frac{1}{x \\ln x}+\\frac{\\sec x}{\\sin x}+1\\right] \\\\\n\n\\end{array}<\/span>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><strong>\u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u0983<\/strong><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u0995\u09cb\u09a8\u09cb<\/span> <span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=Rkj9BuBoCjc&amp;list=PL0dr4HGr8HPiPsV8-EEQp7jXqdnKyjhFg\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">\u09ab\u09be\u0982\u09b6\u09be\u09a8<\/span><\/a><\/span> <span style=\"font-weight: 400;\">\u09af\u09be\u0995\u09c7<\/span><span style=\"font-weight: 400;\"> \u00a0 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">y <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\"> f(x)<\/span><\/span><span style=\"font-weight: 400;\"> \u09ac\u09be <span class=\"katex-eq\" data-katex-display=\"false\"> y<\/span> \u0995\u09c7\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09ac\u09be <span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span> \u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\">y <\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df, \u09a4\u09be \u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09ae\u09a8:\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">y <\/span> <\/span><span style=\"font-weight: 400;\">= <span class=\"katex-eq\" data-katex-display=\"false\">3 x^{2} <\/span> <\/span><span style=\"font-weight: 400;\">+ <span class=\"katex-eq\" data-katex-display=\"false\"> 2x<\/span> + 1\u00a0<\/span><\/p>\n<h3><span style=\"color: #800080;\"><strong>\u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u0983<\/strong><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7 \u09ab\u09be\u0982\u09b6\u09a8 \u098f <span class=\"katex-eq\" data-katex-display=\"false\">y <\/span> \u0995\u09c7\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09ac\u09be <span class=\"katex-eq\" data-katex-display=\"false\">x <\/span> \u0995\u09c7<span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df \u09a8\u09be, \u09a4\u09be\u0995\u09c7 \u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8 \u09ac\u09b2\u09c7<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u09af\u09c7\u09ae\u09a8:\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> x^{2}<\/span><\/span><span style=\"font-weight: 400;\">&#8211;<span class=\"katex-eq\" data-katex-display=\"false\"> 3 x y<\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> 7 y^{2} + 3<\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">\u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u0995\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> f(x,y) <\/span><\/span><strong>=<\/strong>0<span style=\"font-weight: 400;\">\u00a0\u0986\u0995\u09be\u09b0\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09a4\u09c7 \u09b9\u09df<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/li>\n<\/ul>\n<h3><span style=\"color: #800080;\"><strong>\u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u0983<\/strong><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u0985\u09ac\u09cd\u09af\u0995\u09cd\u09a4<\/span> <span style=\"font-weight: 400;\">\u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0<\/span> <span style=\"font-weight: 400;\">\u0985\u09a8\u09cd\u09a4\u09b0\u099c<\/span> <span style=\"font-weight: 400;\">\u09a8\u09bf\u09b0\u09cd\u09a3\u09df<\/span> <span style=\"font-weight: 400;\">\u0995\u09b0\u09a4\u09c7<\/span><span style=\"font-weight: 400;\">\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span> <\/span><span style=\"font-weight: 400;\">\u0995\u09c7<\/span> <span style=\"font-weight: 400;\">\u09aa\u09b0\u09bf\u09ac\u09b0\u09cd\u09a4\u09a8\u09b6\u09c0\u09b2<\/span> <span style=\"font-weight: 400;\">\u098f\u09ac\u0982 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> y <\/span> <\/span><span style=\"font-weight: 400;\">\u0995\u09c7<\/span><span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">x <\/span> <\/span><span style=\"font-weight: 400;\">\u098f\u09b0<\/span> <span style=\"font-weight: 400;\">\u09ab\u09be\u0982\u09b6\u09a8<\/span> <span style=\"font-weight: 400;\">\u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be<\/span> <span style=\"font-weight: 400;\">\u0995\u09b0\u09c7<\/span> <span style=\"font-weight: 400;\">\u09aa\u09cd\u09b0\u09a4\u09cd\u09af\u09c7\u0995<\/span> <span style=\"font-weight: 400;\">\u09aa\u09a6\u0995\u09c7<\/span> <span style=\"font-weight: 400;\">\u09aa\u09c3\u09a5\u0995<\/span> <span style=\"font-weight: 400;\">\u09aa\u09c3\u09a5\u0995\u09ad\u09be\u09ac\u09c7<\/span> <span style=\"font-weight: 400;\">\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3<\/span> <span style=\"font-weight: 400;\">\u0995\u09b0\u09c7<\/span> <span style=\"font-weight: 400;\"> <span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u09a4\u09c7 \u09b9\u09df<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Q. <span class=\"katex-eq\" data-katex-display=\"false\"> x^{3}<\/span>&#8211;<span class=\"katex-eq\" data-katex-display=\"false\"> 3 x^{2} y<\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> y^{3} <\/span>=5<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u21d2<span class=\"katex-eq\" data-katex-display=\"false\"> x^{3}<\/span>&#8211;<span class=\"katex-eq\" data-katex-display=\"false\"> 3 x^{2} y<\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> y^{3} <\/span><\/span><span style=\"font-weight: 400;\">-5=0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u21d2<\/span><span style=\"font-weight: 400;\">3<span class=\"katex-eq\" data-katex-display=\"false\"> x^{2}<\/span><\/span><span style=\"font-weight: 400;\">-3(<\/span><span style=\"font-weight: 400;\">y . <span class=\"katex-eq\" data-katex-display=\"false\">2x <\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> x^{2}<\/span><\/span><span style=\"font-weight: 400;\">.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span>)<\/span><span style=\"font-weight: 400;\">+3<span class=\"katex-eq\" data-katex-display=\"false\"> y^{2}<\/span><\/span><span style=\"font-weight: 400;\">.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u21d2<span class=\"katex-eq\" data-katex-display=\"false\"> x^{3}<\/span><\/span><span style=\"font-weight: 400;\">&#8211;<span class=\"katex-eq\" data-katex-display=\"false\"> 6cy <\/span>-3<span class=\"katex-eq\" data-katex-display=\"false\"> x^{2}<\/span>.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">+3<span class=\"katex-eq\" data-katex-display=\"false\"> y^{2}<\/span>.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u21d2<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{\\left(3 x^{2}-3 y^{2}\\right) d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">3 x^{2} <\/span><\/span><span style=\"font-weight: 400;\">&#8211;<span class=\"katex-eq\" data-katex-display=\"false\"> 6 x y<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u21d2 <span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{3 x^{3}-6 x y}{3 x^{2}-3 y^{2}} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">(Ans)<\/span><\/p>\n<p><b>Shortcut:<\/b><\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-9350 size-full\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/02\/1.png\" alt=\"\u09ab\u09be\u0982\u09b6\u09a8 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7\" width=\"1015\" height=\"217\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/02\/1.png 1015w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/02\/1-300x64.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/02\/1-768x164.png 768w\" sizes=\"(max-width: 1015px) 100vw, 1015px\" \/><\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{l}\n\n\\text { Q. } \\boldsymbol{x}^{3}-\\mathbf{3} \\boldsymbol{x}^{2} \\boldsymbol{y}-\\boldsymbol{y}^{3}-\\mathbf{5}=\\mathbf{0} \\\\\n\n\\frac{d y}{d x}=-\\frac{\\frac{d}{d x}^{\\left(x^{3}-3 x^{2} y+y^{3}-5\\right)}}{\\frac{d}{d y}^{\\left(x^{3}-3 x^{2} y+y^{3}-5\\right)}} \\\\\n\n=\\frac{3 x^{2}-6 x y+0-0}{0-3 x^{2}+3 y^{2}-0} \\\\\n\n=\\frac{3 x^{2}-6 x y}{3 y^{2}+3 y^{2}} \\\\\n\n=\\frac{3 x^{2}-6 x y}{3 x^{2}+3 y^{2}} \\\\\n\n\\text { (Ans) }\n\n\\end{array}<\/span>\n<p>&nbsp;<\/p>\n<p><b>\u0989\u09a6\u09be\u09b9\u09b0\u09a8-8: <\/b><strong><span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span>\u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> x^{y}<\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">y^{x} <\/span>\u00a0\u098f\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0<\/strong><b>\u0964<\/b><\/p>\n<p><b>\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 <\/b><span style=\"font-weight: 400;\">(a) \u09a6\u09c7\u0993\u09df\u09be \u0986\u099b\u09c7, <span class=\"katex-eq\" data-katex-display=\"false\">x^{y}=y^{x}<\/span>\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> X <\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0989\u09ad\u09df \u09aa\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a8 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{l}\n\n\\left.x^{y}\\left[y \\frac{d y}{d x}(\\ln y)+\\ln \\mathrm{x} \\frac{\\mathrm{d} y}{\\mathrm{dx}}\\right)\\right]=\\mathrm{y}^{\\mathrm{x}}\\left[\\mathrm{x} \\frac{\\mathrm{d}}{\\mathrm{dx}}(\\ln y)+\\ln y \\frac{\\mathrm{d}}{\\mathrm{dx}}(\\mathrm{x})\\right] \\\\\n\n\\Rightarrow y \\frac{1}{x}+\\ln x \\frac{d y}{d x}=x \\frac{1}{y} \\frac{d y}{d x}+\\ln y\\left[\\therefore x^{y}=y^{x}\\right] \\\\\n\n\\Rightarrow\\left(\\ln x-\\frac{x}{y}\\right) \\cdot \\frac{d y}{d x}=\\ln y-\\frac{y}{x} \\Rightarrow\\left(\\frac{y \\ln x-x}{y}\\right) \\frac{d y}{d x}=\\frac{x \\ln y-y}{x}\n\n\\end{array}<\/span>\n<p>&nbsp;<\/p>\n<span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{l}\n\n\\therefore \\frac{d y}{d x}=\\frac{y(x \\ln y-y)}{x(y \\ln x-x)} \\\\\n\n\\text { (Ans) }\n\n\\end{array}<\/span>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #800080;\"><b>\u09aa\u09b0\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c <\/b><b>(The essence of parametric functions)<\/b><b> :<\/b><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">\u0995\u0996\u09a8 \u0995\u0996\u09a8 \u09b8\u09c1\u09ac\u09bf\u09a7\u09be\u09b0 \u099c\u09a8\u09cd\u09af \u09b8\u099e\u09cd\u099a\u09be\u09b0\u09aa\u09a5\u09c7\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a8\u09c7 <span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span> \u0993\u00a0 <span class=\"katex-eq\" data-katex-display=\"false\">y <\/span> \u099a\u09b2\u0995 \u09b0\u09be\u09b6\u09bf\u0995\u09c7 \u09a4\u09c3\u09a4\u09c0\u09df \u0986\u09b0\u09c7\u0995\u099f\u09bf \u099a\u09b2\u09b0\u09be\u09b6\u09bf\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df<\/span><span style=\"font-weight: 400;\">\u0964<\/span> <span style=\"font-weight: 400;\">\u0990 \u09a4\u09c3\u09a4\u09c0\u09df \u099a\u09b2\u09b0\u09be\u09b6\u09bf \u09b9\u09b2\u09cb \u09aa\u09b0\u09bf\u09ae\u09bf\u09a4\u09bf<\/span><span style=\"font-weight: 400;\">\u0964<\/span><span style=\"font-weight: 400;\"> \u0986\u09b0 \u09b8\u09c0\u09ae\u0995\u09b0\u09a3\u099f\u09bf \u09aa\u09b0\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<\/span><span style=\"font-weight: 400;\">\u0964<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">y <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\"> g(t) <\/span><\/span><span style=\"font-weight: 400;\">\u098f\u09ac\u0982<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">f(t) <\/span><\/span><span style=\"font-weight: 400;\"> \u09b9\u09b2\u09c7 &#8211;\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d t} <\/span><\/span><span style=\"font-weight: 400;\">.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d t}{d x} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d t} <\/span> x <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\frac{1}{d x}}{d t}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d t} <\/span><\/span><span style=\"font-weight: 400;\">\/<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d x}{d t} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">Q. y<span class=\"katex-eq\" data-katex-display=\"false\">\\left.\\frac{y=\\cos \\theta}{x=a \\sin \\theta}\\right\\} <\/span><\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span>=<\/span><span style=\"font-weight: 400;\">?<\/span><\/p>\n<p><b>\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-\u09e7:<\/b><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d \\theta} <\/span><\/span><span style=\"font-weight: 400;\"> \/ <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d x}{d \\theta}<\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^{\\operatorname{acos} \\theta}}{d \\theta} <\/span><\/span><span style=\"font-weight: 400;\">\/<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^{\\operatorname{asin} \\theta}}{d \\theta} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">-\\operatorname{asin} \\theta <\/span>\/<span class=\"katex-eq\" data-katex-display=\"false\">a \\cos \\theta <\/span><\/span>\u00a0<span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">=-<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\tan \\theta<\/span><\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(Ans)<\/span><\/p>\n<p><b>\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-2:<\/b><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> y<\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\operatorname{acos} \\theta <\/span>;\u2234<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\cos \\theta <\/span>=<\/span><span style=\"font-weight: 400;\">y<\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{y}{a}<\/span> <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">x <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\operatorname{asin} \\theta <\/span>;\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\u00a0\\sin \\theta <\/span>=<\/span><span style=\"font-weight: 400;\">\u00a0 <\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{x}{a} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<span class=\"katex-eq\" data-katex-display=\"false\">\\cos ^{2} \\theta <\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> \\sin ^{2} \\theta<\/span>=1<\/span><span style=\"font-weight: 400;\">\u00a0 <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{y^{2}}{a^{2}} <\/span><\/span><span style=\"font-weight: 400;\">+<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{x^{2}}{a^{2}} <\/span><\/span><span style=\"font-weight: 400;\">=1<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\"> x^2<\/span><\/span><span style=\"font-weight: 400;\">+<span class=\"katex-eq\" data-katex-display=\"false\"> y^{2}<\/span><\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">a^{2} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\"> x^2<\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> y^{2}<\/span><\/span><span style=\"font-weight: 400;\">&#8211;<span class=\"katex-eq\" data-katex-display=\"false\">a^{2} <\/span>\u00a0<\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d^{x^{2}}}{d x}<\/span><\/span><span style=\"font-weight: 400;\">+<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^{y^{2}}}{d x} <\/span><\/span><span style=\"font-weight: 400;\">&#8211;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d^{a^{2}}}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">2x <\/span>+<span class=\"katex-eq\" data-katex-display=\"false\"> 2 y<\/span>.<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d^{y}}{d x}<\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d x}^{y} <\/span><\/span><span style=\"font-weight: 400;\">=-<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{2 x}{2 y} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=-<\/span> <span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{\\operatorname{asin} \\theta}{\\operatorname{acos} \\theta}<\/span><span style=\"font-weight: 400;\">=-<span class=\"katex-eq\" data-katex-display=\"false\"> \\tan \\theta<\/span> <\/span>\u00a0<span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(Ans)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Q. <span class=\"katex-eq\" data-katex-display=\"false\"> \\left.\\frac{x=a t^{2}}{y=2 a t}\\right\\} \\frac{d y}{d x}<\/span>=?<\/span><\/p>\n<p><strong>\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-\u09e7:<\/strong><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d}{d x}^{y}<\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d t}^{y} <\/span><\/span><span style=\"font-weight: 400;\">\/<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d}{d t}^{x} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d^{2 a t}}{d t}<\/span><\/span><span style=\"font-weight: 400;\">\/<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d^{a t^{2}}}{d t}<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">2a <\/span><\/span><span style=\"font-weight: 400;\">\/<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">2at <\/span><\/span> <span style=\"font-weight: 400;\">=<\/span><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\frac{1}{t}\u00a0<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">(Ans)<\/span><\/p>\n<p><strong>\u09aa\u09a6\u09cd\u09a7\u09a4\u09bf-2:<\/strong><\/p>\n<p><span style=\"font-weight: 400;\"><span class=\"katex-eq\" data-katex-display=\"false\">y <\/span>=<\/span><span style=\"font-weight: 400;\">\u00a0<span class=\"katex-eq\" data-katex-display=\"false\"> 2at<\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">t <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{y}{2 a} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u2234<span class=\"katex-eq\" data-katex-display=\"false\">x <\/span>=<span class=\"katex-eq\" data-katex-display=\"false\">a <\/span><span class=\"katex-eq\" data-katex-display=\"false\">\\left(\\frac{y}{2 a}\\right)^{2} <\/span><\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\"> x<\/span>&#8211;<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{y^{2}}{4 a} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9-<span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{2 y}{4 a}<\/span><\/span><span style=\"font-weight: 400;\">.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{y}{2 a} <\/span><\/span><span style=\"font-weight: 400;\">.<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{y}{2 a} <\/span><span class=\"katex-eq\" data-katex-display=\"false\"> \\frac{d y}{d x}<\/span><\/span><span style=\"font-weight: 400;\">=0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u27f9<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{d y}{d x} <\/span><\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{2 a}{y} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{2 a}{2 a t} <\/span><\/span><span style=\"font-weight: 400;\">=<span class=\"katex-eq\" data-katex-display=\"false\">\\frac{1}{t} <\/span><\/span><span style=\"font-weight: 400;\">\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">(Ans)<\/span><\/p>\n<div class=\"x1tlxs6b x1g8br2z x1gn5b1j x230xth x14ctfv x1okitfd x6ikm8r x10wlt62 x1mzt3pk x1y1aw1k xn6708d xwib8y2 x1ye3gou x1n2onr6 x13faqbe x1vjfegm\" role=\"none\">\n<div class=\"\">\n<div class=\"x9f619 x1n2onr6 x1ja2u2z __fb-light-mode\" role=\"none\">\n<hr \/>\n<p dir=\"auto\" role=\"none\">\n<p class=\"x6prxxf x1fc57z9 x1yc453h x126k92a xzsf02u\" dir=\"auto\" role=\"none\"><em><strong>\u098f\u0987\u099a\u098f\u09b8\u09b8\u09bf \u0993 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u09aa\u09b0\u09c0\u0995\u09cd\u09b7\u09be\u09b0\u09cd\u09a5\u09c0\u09a6\u09c7\u09b0 \u099c\u09a8\u09cd\u09af \u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<ul>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-25-online-batch-2-bangla-english-ict\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a \u09e8.\u09e6 (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-26-online-batch-bangla-english-ict\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ac\u09be\u0982\u09b2\u09be, \u0987\u0982\u09b0\u09c7\u099c\u09bf, \u09a4\u09a5\u09cd\u09af \u0993 \u09af\u09cb\u0997\u09be\u09af\u09cb\u0997 \u09aa\u09cd\u09b0\u09af\u09c1\u0995\u09cd\u09a4\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2025-online-batch\/\">HSC 25 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/hsc-2026-online-batch\/\">HSC 26 \u0985\u09a8\u09b2\u09be\u0987\u09a8 \u09ac\u09cd\u09af\u09be\u099a (\u09ab\u09bf\u099c\u09bf\u0995\u09cd\u09b8, \u0995\u09c7\u09ae\u09bf\u09b8\u09cd\u099f\u09cd\u09b0\u09bf, \u09ae\u09cd\u09af\u09be\u09a5, \u09ac\u09be\u09df\u09cb\u09b2\u099c\u09bf)<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/medical-admission-course\/\">\u09ae\u09c7\u09a1\u09bf\u0995\u09c7\u09b2 \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-a-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/dhaka-university-b-unit-admission-course\/\">\u09a2\u09be\u0995\u09be \u09ad\u09be\u09b0\u09cd\u09b8\u09bf\u099f\u09bf B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/buet-ques-solve\/\">\u09ac\u09c1\u09df\u09c7\u099f \u0995\u09cb\u09b6\u09cd\u099a\u09c7\u09a8 \u09b8\u09b2\u09ad \u0995\u09cb\u09b0\u09cd\u09b8<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-a-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b A Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<li role=\"none\"><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/gst-b-unit-admission-course\/\">\u0997\u09c1\u099a\u09cd\u099b B Unit \u098f\u09a1\u09ae\u09bf\u09b6\u09a8 \u0995\u09cb\u09b0\u09cd\u09b8 &#8211; \u09e8\u09e6\u09e8\u09ea<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p><em><strong>\u0986\u09ae\u09be\u09a6\u09c7\u09b0 \u09b8\u09cd\u0995\u09bf\u09b2 \u09a1\u09c7\u09ad\u09c7\u09b2\u09aa\u09ae\u09c7\u09a8\u09cd\u099f \u0995\u09cb\u09b0\u09cd\u09b8\u09b8\u09ae\u09c2\u09b9\u0983<\/strong><\/em><\/p>\n<ul>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/study-abroad-complete-guideline\/\">\u09ac\u09bf\u09a6\u09c7\u09b6\u09c7 \u0989\u099a\u09cd\u099a\u09b6\u09bf\u0995\u09cd\u09b7\u09be: Study Abroad Complete Guideline<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/student-hacks\/\">Student Hacks<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ielts-course\/\">IELTS Course by Munzereen Shahid<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/english-grammar-course\/\">Complete English Grammar Course<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ms-bundle\/\"> Microsoft Office 3 in 1 Bundle<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/ghore-boshe-freelancing\/\">\u0998\u09b0\u09c7 \u09ac\u09b8\u09c7 Freelancing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/facebook-marketing\/\">Facebook Marketing<\/a><\/span><\/li>\n<li><span style=\"color: #0000ff;\"><a style=\"color: #0000ff;\" href=\"https:\/\/10minuteschool.com\/product\/adobe-4-in-1-bundle\/\">Adobe 4 in 1 Bundle<\/a><\/span><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><em>\u09e7<\/em><em>\u09e6 \u09ae\u09bf\u09a8\u09bf\u099f \u09b8\u09cd\u0995\u09c1\u09b2\u09c7\u09b0 \u0995\u09cd\u09b2\u09be\u09b8\u0997\u09c1\u09b2\u09cb \u0985\u09a8\u09c1\u09b8\u09b0\u09a3 \u0995\u09b0\u09a4\u09c7 \u09ad\u09bf\u099c\u09bf\u099f: <span style=\"color: #993300;\"><strong><a style=\"color: #993300;\" href=\"https:\/\/10minuteschool.com\/?ref=https%3A%2F%2Fblog.10minuteschool.com%2Fwordpress%2F&amp;post_id=78178&amp;blog_category_id=700\">www.10minuteschool.com<\/a><\/strong><\/span><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u09ab\u09be\u0982\u09b6\u09a8 \u0995\u09be\u0995\u09c7 \u09ac\u09b2\u09c7? \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09b2 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u09b8\u09c7\u099f\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u099f\u09bf \u0989\u09aa\u09be\u09a6\u09be\u09a8\u0995\u09c7 \u0985\u09a8\u09cd\u09af \u098f\u0995\u099f\u09bf \u09b8\u09c7\u099f\u09c7\u09b0 \u098f\u0995\u0995 \u0989\u09aa\u09be\u09a6\u09be\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u09b8\u09ae\u09cd\u09aa\u09b0\u09cd\u0995\u09bf\u09a4 \u0995\u09b0\u09c7 \u098f\u09ae\u09a8 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09af\u09bc\u09ae\u0964 \u09b8\u09b9\u099c \u0995\u09a5\u09be\u09df, \u09ab\u09be\u0982\u09b6\u09a8 \u09b9\u09b2 \u098f\u0995 \u09a7\u09b0\u09a8\u09c7\u09b0 \u09af\u09a8\u09cd\u09a4\u09cd\u09b0 \u09af\u09be \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0987\u09a8\u09aa\u09c1\u099f \u09a6\u09bf\u09b2\u09c7 \u098f\u0995\u099f\u09bf \u09a8\u09bf\u09b0\u09cd\u09a6\u09bf\u09b7\u09cd\u099f \u0986\u0989\u099f\u09aa\u09c1\u099f \u09a6\u09c7\u09df\u0964 \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u099c\u00a0\u09a8\u09bf\u09b0\u09cd\u09a3\u09df (Determining<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/derivative-of-different-functions\/\">Read More<\/a><\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[3026,4264,3037],"tags":[2937,2946,1877,99,2944,2942,2707,2945],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4311"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=4311"}],"version-history":[{"count":24,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4311\/revisions"}],"predecessor-version":[{"id":16133,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/4311\/revisions\/16133"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=4311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=4311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=4311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}