{"id":5577,"date":"2021-09-21T17:03:35","date_gmt":"2021-09-21T11:33:35","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5577"},"modified":"2021-11-15T16:09:49","modified_gmt":"2021-11-15T10:39:49","slug":"higher-order-derivatives-of-parametric-equations","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/higher-order-derivatives-of-parametric-equations\/","title":{"rendered":"Higher Order Derivatives of Parametric Equations"},"content":{"rendered":"

Here you will learn higher order derivatives of parametric equations with examples.<\/p>\n

Let’s begin –<\/p>\n

Higher Order Derivatives of Parametric Equations<\/h2>\n

We know that the differentiation of parametric equations of type x = at and y = 2at is given by formula<\/p>\n

\n

\\(dy\\over dx\\) = \\(dy\/dt\\over dx\/dt\\)<\/p>\n

where t is the parameter<\/p>\n<\/blockquote>\n

Now, by again differentiating the above equation we obtain 2nd order derivative given below:<\/p>\n

\n

\\(d^2y\\over dx^2\\) = \\(d\\over dx\\)(\\(dy\\over dx\\))<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : find \\(d^2y\\over dx^2\\) , if x = \\(at^2\\) , y = 2at.<\/p>\n

Solution<\/strong><\/span> : We have, <\/p>\n

x = \\(at^2\\) , y = 2at.<\/p>\n

Differentiating both sides with respect to t,<\/p>\n

\\(\\implies\\) \\(dx\\over dt\\) = 2at and \\(dy\\over dt\\) = 2a           …….(i)<\/p>\n

\\(\\therefore\\) \\(dy\\over dx\\) = \\(dy\/dt\\over dx\/dt\\) = \\(2a\\over 2at\\) = \\(1\\over t\\)<\/p>\n

Differentiating both sides with respect to x, we get<\/p>\n

\\(d^2y\\over dx^2\\) = \\(d\\over dx\\)(\\(1\\over t\\))<\/p>\n

\\(\\implies\\) \\(d^2y\\over dx^2\\) = \\(-1\\over t^2\\)\\(dt\\over dx\\)<\/p>\n

from (i), [\\(dx\\over dt\\) = 2at \\(\\therefore\\)  \\(dt\\over dx\\) = \\(1\\over 2at\\)]<\/p>\n

\\(\\implies\\) \\(d^2y\\over dx^2\\)  = -\\(1\\over 2at^3\\)<\/p>\n

Example<\/strong><\/span> : If x = \\(acos^3\\theta\\) , y = \\(asin^3\\theta\\) , find \\(d^2y\\over dx^2\\). Also find its value at \\(\\theta\\) = \\(\\pi\\over 6\\).<\/p>\n

Solution<\/strong><\/span> : We have, <\/p>\n

x = \\(acos^3\\theta\\) and y = \\(asin^3\\theta\\)<\/p>\n

Differentiating both sides with respect to \\(\\theta\\), <\/p>\n

\\(\\therefore\\) \\(dx\\over d\\theta\\) = \\(-3acos^2\\theta sin\\theta\\) and \\(dy\\over d\\theta\\) = \\(3asin^2\\theta cos\\theta\\)               …….(i)<\/p>\n

 So, \\(dy\\over dx\\) = \\(dy\/d\\theta\\over dx\/d\\theta\\) = \\(3asin^2\\theta cos\\theta\\over {-3acos^2\\theta sin\\theta}\\) = \\(-tan\\theta\\)<\/p>\n

Differentiating both sides with respect to x, we obtain<\/p>\n

\\(d^2y\\over dx^2\\) = \\(d\\over dx\\)(\\(-tan\\theta\\)) <\/p>\n

= \\(sec^2\\theta\\)\\(d\\theta\\over dx\\) <\/p>\n

from (i), [\\(dx\\over d\\theta\\) = \\(-3acos^2\\theta sin\\theta\\) \\(\\therefore\\)  \\(d\\theta\\over dx\\) = \\(1\\over {-3acos^2\\theta sin\\theta}\\)]<\/p>\n

\\(d^2y\\over dx^2\\) = -\\(sec^2\\theta\\)\\(\\times\\)\\(1\\over {-3acos^2\\theta sin\\theta}\\)<\/p>\n

\\(\\implies\\) \\(d^2y\\over dx^2\\) = \\(1\\over 3a\\) \\(sec^4\\theta\\)\\(cosec\\theta\\)<\/p>\n

\\(\\therefore\\) At \\(\\theta\\) = \\(\\pi\\over 6\\) , \\(d^2y\\over dx^2\\)  = \\(1\\over 3a\\) \\(sec^4{\\pi\\over 4}\\)\\(cosec{\\pi\\over 6}\\)<\/p>\n

= \\(1\\over 3a\\) \\(\\times\\) \\((2\\over \\sqrt{3})^4\\) \\(\\times\\) 2 = \\(32\\over 27a\\) <\/p>\n\n\n

\n
Next – Derivative as a Rate Measure Class 12<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Higher Order Derivatives – Definition & Example<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn higher order derivatives of parametric equations with examples. Let’s begin – Higher Order Derivatives of Parametric Equations We know that the differentiation of parametric equations of type x = at and y = 2at is given by formula \\(dy\\over dx\\) = \\(dy\/dt\\over dx\/dt\\) where t is the parameter Now, by again …<\/p>\n

Higher Order Derivatives of Parametric Equations<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[37],"tags":[76,77,122,121],"yoast_head":"\nHigher Order Derivatives of Parametric Equations - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn higher order derivatives of parametric equations with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/higher-order-derivatives-of-parametric-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Higher Order Derivatives of Parametric Equations - 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