{"id":4314,"date":"2024-01-30T12:22:03","date_gmt":"2024-01-30T06:22:03","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=4314"},"modified":"2024-11-05T16:02:56","modified_gmt":"2024-11-05T10:02:56","slug":"%e0%a6%85%e0%a6%b8%e0%a7%80%e0%a6%ae-%e0%a6%ab%e0%a6%be%e0%a6%82%e0%a6%b6%e0%a6%a8%e0%a7%87%e0%a6%b0-%e0%a6%85%e0%a6%a8%e0%a7%8d%e0%a6%a4%e0%a6%b0%e0%a7%80%e0%a6%95%e0%a6%b0%e0%a6%a3","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/%e0%a6%85%e0%a6%b8%e0%a7%80%e0%a6%ae-%e0%a6%ab%e0%a6%be%e0%a6%82%e0%a6%b6%e0%a6%a8%e0%a7%87%e0%a6%b0-%e0%a6%85%e0%a6%a8%e0%a7%8d%e0%a6%a4%e0%a6%b0%e0%a7%80%e0%a6%95%e0%a6%b0%e0%a6%a3\/","title":{"rendered":"\u0985\u09b8\u09c0\u09ae \u09aa\u09b0\u09cd\u09af\u09a8\u09cd\u09a4 \u099a\u09b2\u09a4\u09c7 \u09a5\u09be\u0995\u09be \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3\u00a0(Differentiation of a function that lasts indefinitely)"},"content":{"rendered":"\r\n\r\n\r\n

\u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3: <\/strong><\/span>\u09aa\u09b0\u09cd\u09af\u09be\u09df\u0995\u09cd\u09b0\u09ae\u09bf\u0995<\/b> \u0985\u09a8\u09cd\u09a4\u09b0\u099c<\/b><\/span><\/h3>\r\n

\r\n\r\n<\/b><\/span><\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
\u00a0<\/td>\r\nY<\/b><\/td>\r\nf(x)<\/strong><\/span><\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n
\\frac{d y}{d x} <\/span><\/td>\r\ny_{1} <\/span><\/td>\r\n\\boldsymbol{f}^{\\prime}(\\boldsymbol{x}) <\/span><\/td>\r\n\\frac{d y}{d x} <\/span><\/td>\r\n<\/tr>\r\n
\\frac{d^{2} y}{d x^{2}} <\/span><\/td>\r\ny_{2} <\/span><\/td>\r\n \\boldsymbol{f}^{\\prime \\prime}(\\boldsymbol{x})<\/span><\/td>\r\n\\frac{d}{d x} <\/span> ( \\frac{d y}{d x} <\/span> )<\/span><\/td>\r\n<\/tr>\r\n
\\frac{d^{3} y}{d x^{3}} <\/span><\/td>\r\ny_{3} <\/span><\/td>\r\n\\boldsymbol{f}^{\\prime \\prime \\prime}(\\boldsymbol{x}) <\/span><\/td>\r\n\\frac{d}{d x} <\/span> ( \\frac{d^{3} y}{d x^{3}} <\/span> )<\/span><\/td>\r\n<\/tr>\r\n
\\frac{d^{4} y}{d x^{4}} <\/span><\/td>\r\ny_{4} <\/span><\/td>\r\n\\boldsymbol{f}^{\\prime \\prime \\prime \\prime}(\\boldsymbol{x}) <\/span><\/td>\r\n\\frac{d}{d x} <\/span> ( \\frac{d^{4} y}{d x^{4}} <\/span> )<\/span><\/td>\r\n<\/tr>\r\n
*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n

*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n

*<\/b>\r\n

 <\/p>\r\n

*<\/b><\/p>\r\n<\/td>\r\n

*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n

*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n

*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n

*<\/b>\r\n

 <\/p>\r\n

*<\/b><\/p>\r\n<\/td>\r\n

*<\/span>\r\n

 <\/p>\r\n

*<\/span><\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/figure>\r\n

\r\n\r\n<\/b><\/span><\/h3>\r\n

\u098f\u09ad\u09be\u09ac\u09c7 \u0995\u09cb\u09a8\u09cb \u09ab\u09be\u0982\u09b6\u09a8\u09c7\u09b0 \u09a7\u09be\u09b0\u09be\u09ac\u09be\u09b9\u09bf\u0995\u09ad\u09be\u09ac\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09b2\u09c7<\/strong>\u00a0\u09aa\u09cd\u09b0\u0995\u09cd\u09b0\u09bf\u09df\u0995\u09c7 \u09aa\u09b0\u09cd\u09af\u09be\u09df\u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3<\/strong> \u09ac\u09b2\u09c7<\/span>\u0964<\/span><\/p>\r\n

Type-23:\u00a0<\/b><\/span><\/h3>\r\n\r\n\r\n\r\n

Concept: (i) power \u0986\u0995\u09be\u09b0\u09c7 \u09a5\u09be\u0995\u09b2\u09c7 \u098f\u0996\u09be\u09a8\u09c7 \\ln <\/span> \u09a8\u09bf\u09a4\u09c7 \u09b9\u09ac\u09c7<\/span>\u0964<\/span> (ii) Root \u09a5\u09be\u0995\u09b2\u09c7 \u09ac\u09b0\u09cd\u0997 \u0995\u09b0\u09c7 \u0986\u0997\u09be\u09a4\u09c7 \u09b9\u09ac\u09c7<\/span>\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

Example-76. <\/b> y <\/span>=x^{x^{x^{-^{-^{-^{\\infty}}}}}}\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\\mathrm{Sol}^{n} <\/span>:\u00a0 y <\/span> = x^{x^{x^{-^{-^{-^{\\infty}}}}}}\u00a0<\/span>\u27f9ln y <\/span> = ln <\/span>x^{x^{x^{-^{-^{-^{\\infty}}}}}} <\/span> ln x<\/span>\u00a0 \u27f9 ln y<\/span> = yln x <\/span><\/p>\r\n\r\n\r\n\r\n

Differentiate<\/a><\/span> \u0995\u09b0\u09c7 \u09aa\u09be\u0987 \\frac{1}{y}<\/span>\\frac{d y}{d x} <\/span><\/span>=y <\/span>.\\frac{1}{x} <\/span><\/span>+ ln x <\/span> \\frac{d y}{d x} <\/span><\/span>;<\/span> \\frac{d y}{d x} <\/span><\/span>(\\frac{1}{y} <\/span><\/span>–\\ln x\u00a0<\/span><\/span>)<\/span> =\\frac{y}{x} <\/span><\/span>\u00a0\u27f9 \\frac{d y}{d x}<\/span><\/span>=\\frac{y^{2}}{x(1-y \\ln x)} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

Example-77.y <\/span><\/b>= \\sqrt{\\sin x+\\sqrt{\\sin x+\\sqrt{\\sin x+\\ldots \\ldots \\ldots \\infty}}}<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

\\mathrm{Sol}^{n} <\/span>:<\/span>y= \\sqrt{\\sin x+\\sqrt{\\sin x+\\sqrt{\\sin x+\\ldots \\ldots \\ldots \\infty}}}<\/span><\/span>\u00a0<\/span> \u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9 y<\/span>=<\/span> sin x<\/span>+ \\sqrt{\\sin x+\\sqrt{\\sin x+\\sqrt{\\sin x+\\ldots \\ldots \\ldots \\infty}}}<\/span><\/span>\u00a0\u00a0<\/span> \u27f9y^{2} <\/span><\/span>=<\/span> sin x<\/span>+y <\/span><\/p>\r\n\r\n\r\n\r\n

\u2234<\/span> Differentiate<\/a><\/span> \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/span><\/p>\r\n\r\n\r\n\r\n

2y <\/span>\\frac{d y}{d x} <\/span><\/span>= cos x<\/span><\/span>+\\frac{d y}{d x} <\/span><\/span>\u27f9<\/span> \\frac{d y}{d x} <\/span><\/span>(2y-1) <\/span>= cos x<\/span><\/span>\u27f9\\frac{d y}{d x} <\/span><\/span>= \\frac{\\cos x}{2 y-1}<\/span><\/span>\u00a0\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n\r\n\r\n

\u09aa\u09b0\u09cd\u09af\u09be\u09df\u0995\u09cd\u09b0\u09ae\u09bf\u0995 \u0985\u09a8\u09cd\u09a4\u09b0\u099c\u09c7 n \u09a4\u09ae \u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df\u0983\u00a0<\/strong><\/span><\/h3>\r\n\r\n\r\n\r\n

\\rightarrow<\/span> n \u09a4\u09ae \u0985\u09a8\u09cd\u09a4\u09b0\u0995 \u09ae\u09be\u09a8\u09c7\u00a0 <\/span>common pattern<\/strong> <\/span>\u09ac\u09c7\u09b0 \u0995\u09b0\u09be<\/span>\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\\rightarrow<\/span> \u09e9-\u09ea<\/strong> \u09ac\u09be\u09b0 \u0995\u09b0\u09be\u09b0 \u09aa\u09b0 <\/span>common pattern <\/span>\u0986\u09b8\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

(1). <\/b>y <\/span>=x^{n} <\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{1} <\/span><\/span>=n <\/span> x^{n-1}<\/span><\/span>\u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{2} <\/span><\/span>= n <\/span>(n-1) <\/span>x^{n-2} <\/span><\/span>\u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{3} <\/span><\/span>=n <\/span><\/span> (n-1) <\/span>(n-2) <\/span>x^{n-3} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{4} <\/span><\/span>=n <\/span> (n-1) <\/span>(n-2) <\/span>(n-3) <\/span>x^{n-4} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{m} <\/span><\/span>= n_{p_{m}}<\/span> x^{n-m} <\/span><\/span>\u00a0\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u00a0 n\u2265m <\/span><\/span><\/p>\r\n

 <\/p>\r\n\r\n\r\n\r\n

Q. y y <\/span>= x^{5}<\/span>\u00a0\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{1} <\/span><\/span>=5 x^{4}<\/span>\u00a0 <\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{2} <\/span><\/span>=20 x^{3}<\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{3} <\/span><\/span>=60 x^{2}<\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{4} <\/span><\/span>=5p_{5} <\/span><\/span>\u00a0 x^{5-4}<\/span><\/span>=120 x <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{5} <\/span><\/span>=5p_{5} <\/span><\/span>\u00a0 x^{5-5}<\/span><\/span>=120. x^{0}<\/span><\/span> [ \u09af\u0996\u09a8 n=m <\/span>]<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{6} <\/span><\/span>= \\frac{d^{45}}{d x}<\/span><\/span>= \\frac{d^{120}}{d x}<\/span><\/span>=0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234 m>n<\/span>\u00a0<\/span> \u09b9\u09b2\u09c7 \u09aa\u09b0\u09ac\u09b0\u09cd\u09a4\u09c0 <\/span>y=0 <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

(2). <\/b>y <\/span>= sin x<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{1} <\/span><\/span>=<\/span>cos x <\/span>= \\sin \\left(\\frac{\\pi}{2}+x\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{2} <\/span><\/span>=<\/span> -sin x<\/span>= \\sin \\left(\\frac{2\\pi}{2}+x\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{3} <\/span><\/span>=cos\u00a0x <\/span>= \\sin \\left(\\frac{3\\pi}{2}+x\\right)<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{4} <\/span><\/span>=<\/span> sin x <\/span>=[ \\sin \\left(\\frac{4\\pi}{2}+x\\right)<\/span><\/span>\u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234y_{n} <\/span><\/span>= \\sin \\left(\\frac{n\\pi}{2}+x\\right)<\/span><\/span>\u00a0<\/span><\/p>\r\n

 <\/p>\r\n\r\n\r\n\r\n

Q. y_{1000} <\/span>= ?<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9 y_{1000} <\/span><\/span>=[ \\sin \\left(\\frac{1000\\pi}{2}+x\\right)<\/span><\/span>\u00a0= \\sin (500 \\pi+x)<\/span><\/span>\u00a0=<\/span> sin x <\/span><\/span> \u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

(Ans)<\/span><\/p>\r\n\r\n\r\n\r\n

(3). y <\/span><\/b>=e^{m x} <\/span><\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{1} <\/span><\/span>=e^{m x} <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{2} <\/span><\/span>=m^{2} <\/span>e^{m x} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9y_{3} <\/span><\/span>=m^{3} <\/span>e^{m x} <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u2234y_{n} <\/span>=m^{n} <\/span>e^{m x} <\/span>\u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

(Ans)<\/span><\/p>\r\n

 <\/p>\r\n\r\n\r\n\r\n

\u098a\u09a6\u09be\u09b9\u09b0\u09a3-2: <\/b>y <\/span>=\\left(\\cos ^{-1} x\\right)^{2} <\/span>\u00a0 \u09b9\u09b2\u09c7, \u09a6\u09c7\u0996\u09be\u0993 \u09af\u09c7, \\left(1-x^{2}\\right) <\/span> \\frac{d^{2} y}{d x^{2}}<\/span> – x<\/span>\\frac{d y}{d x} <\/span>=2<\/strong>\u00a0<\/b><\/p>\r\n\r\n\r\n\r\n

\u09aa\u09cd\u09b0\u09ae\u09be\u09a8\u0983<\/b> \u098f\u0996\u09be\u09a8\u09c7, <\/span> y<\/span>=\\left(\\cos ^{-1} x\\right)^{2} <\/span><\/span>\u00a0\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u00a0X <\/span>\u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, \\frac{d y}{d x} <\/span><\/span>=2x <\/span>\\cos ^{-1} x <\/span> \\frac{-1}{\\sqrt{1-x^{2}}} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u27f9 \\sqrt{1-x^{2}}<\/span><\/span>\\frac{d y}{d x} <\/span><\/span>=2 \\cos ^{-1} x<\/span>\u27f9(1-x^{2}) <\/span>(\\frac{d y}{d x})^2 <\/span><\/span>=4 <\/span> (\\cos ^{-1} x)^{2} <\/span><\/span>\u27f9(1-x^{2}) <\/span>(\\frac{d y}{d x})^2 <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

= 4y <\/span><\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u00a0X <\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 \u0985\u09a8\u09cd\u09a4\u09b0\u09c0\u0995\u09b0\u09a3 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, (1-x^{2}) <\/span><\/span>.2\\frac{d y}{d x} <\/span><\/span>. \\frac{d^{2} y}{d x^{2}}<\/span><\/span>+(\\frac{d y}{d x})^2 <\/span>(-2x) <\/span><\/span>=4\\frac{d y}{d x} <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0989\u09ad\u09df \u09aa\u0995\u09cd\u09b7\u0995\u09c7 2\\frac{d y}{d x} <\/span><\/span>\u00a0\u00a0<\/span>\u09a6\u09cd\u09ac\u09be\u09b0\u09be<\/span> \u09ad\u09be\u0997 \u0995\u09b0\u09c7 \u09aa\u09be\u0987, (1-x^{2}) <\/span> \\frac{d^{2} y}{d x^{2}}<\/span><\/span>– x <\/span>\\frac{d y}{d x} <\/span><\/span>=2<\/span>\u00a0<\/span><\/p>\r\n

 <\/p>\r\n\r\n\r\n\r\n

\u098a\u09a6\u09be\u09b9\u09b0\u09a3-3: x<\/span> \u098f\u09b0 \u09b8\u09be\u09aa\u09c7\u0995\u09cd\u09b7\u09c7 <\/b> cos 3x<\/span>\u00a0<\/span> \u098f\u09b0 n <\/span> \u09a4\u09ae \u0985\u09a8\u09cd\u09a4\u09b0\u099c \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0<\/b>\u0964<\/b><\/p>\r\n\r\n\r\n\r\n

\u09b8\u09ae\u09be\u09a7\u09be\u09a8\u0983 <\/b>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, <\/span>y <\/span>= cos 3x<\/span>\u00a0 <\/span>\\therefore y_{1} <\/span><\/span>=-3<\/span> sin 3x<\/span>=3<\/span> \\cos \\left(\\frac{\\pi}{2}+3 x\\right)<\/span><\/span>\u00a0 \u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

y_{2} <\/span> = – 3^{2} <\/span>\\sin \\left(\\frac{\\pi}{2}+3 x\\right) <\/span><\/span>=3^{2} <\/span> \\cos \\left(\\frac{\\pi}{2}+\\frac{\\pi}{2}+3 x\\right) <\/span><\/span>=3^{2} <\/span><\/span> \\cos \\left(2 \\cdot \\frac{\\pi}{2}+3 x\\right) <\/span>\u00a0<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

y_{3} <\/span> = – 3^{3} <\/span>\\sin \\left(2 \\cdot \\frac{\\pi}{2}+3 x\\right) <\/span><\/span>=3^{3} <\/span><\/span>\\cos \\left(\\frac{\\pi}{2}+2 \\cdot \\frac{\\pi}{2}+3 x\\right) <\/span><\/span>=3^{2} <\/span>\\cos \\left(3 \\cdot \\frac{\\pi}{2}+3 x\\right) <\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09a8\u09c1\u09b0\u09c1\u09aa\u09ad\u09be\u09ac\u09c7<\/span>, y_{n} <\/span> <\/span>= – 3^{n} <\/span>\u00a0 <\/span>\\therefore D^{n} <\/span>\u00a0 <\/span>(cos 3x) <\/span>=3^{n} <\/span>\u00a0 \\cos \\left(n \\cdot \\frac{\\pi}{2}+3 x\\right) <\/span>\u00a0\u00a0\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

(Ans)<\/span><\/p>\r\n

\r\n
\r\n
\r\n

\u00a0<\/p>\r\n

\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>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n