{"id":5131,"date":"2021-09-08T20:36:28","date_gmt":"2021-09-08T15:06:28","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5131"},"modified":"2021-11-22T21:41:06","modified_gmt":"2021-11-22T16:11:06","slug":"differentiation-of-implicit-function","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/differentiation-of-implicit-function\/","title":{"rendered":"Differentiation of Implicit Function"},"content":{"rendered":"

Here you will learn what is the differentiation of implicit function with examples.<\/p>\n

Let’s begin –<\/p>\n

Differentiation of Implicit Function<\/h2>\n

If the variables x and y are connected by a relation of the form f(x,y) = 0 and it is not possible or convenient to express y as a function x in the form y\u00a0 = \\(\\phi (x)\\), then y is said to be an implicit function of x. To find \\(dy\\over dx\\) in such a case, we differentiate both sides of the given relation with respect to x, keeping in mind that the derivative of \\(\\phi (y)\\) with respect to x is \\(d\\phi\\over dy\\).\\(dy\\over dx\\).<\/p>\n

for example<\/strong>, \\(d\\over dx\\) (sin y) = cos y. \\(dy\\over dx\\) , \\(d\\over dx\\) (\\(y^2\\)) = 2y\\(dy\\over dx\\)<\/p>\n

It should be noted that \\(d\\over dy\\) (sin y) = cos y but \\(d\\over dx\\) (sin y) = cos y. \\(dy\\over dx\\).<\/p>\n

Similarly , \\(d\\over dy\\) \\((y^3)\\) = \\(3y^2\\) whereas \\(d\\over dx\\) \\((y^3)\\) = \\(3y^2\\) \\(dy\\over dx\\).<\/p>\n

Example<\/strong><\/span> : If \\(ax^2 + 2hxy + by^2 + 2gx + 2fy + c\\) = 0, find \\(dy\\over dx\\).<\/p>\n

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

\\(ax^2 + 2hxy + by^2 + 2gx + 2fy + c\\) = 0<\/p>\n

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

\\(d\\over dx\\) \\((ax^2)\\) + \\(d\\over dx\\)(2hxy) + \\(d\\over dx\\)\\((by^2)\\) + \\(d\\over dx\\)(2gx) + \\(d\\over dx\\)(2fy) + \\(d\\over dx\\)(c) = \\(d\\over dx\\)(0)<\/p>\n

\\(\\implies\\) a\\(d\\over dx\\)\\((x^2)\\) + 2h\\(d\\over dx\\)(xy) + b\\(d\\over dx\\)\\((y^2)\\) + 2g\\(d\\over dx\\)(x) + 2f\\(d\\over dx\\)(y) + 0 = 0<\/p>\n

\\(\\implies\\) 2ax + 2h\\((x{dy\\over dx} + y)\\) + b 2y\\(dy\\over dx\\) + 2g.1 + 2f.\\(dy\\over dx\\) = 0<\/p>\n

= \\(dy\\over dx\\)(2hx + 2by + 2f) + 2ax + 2hy + 2g = 0<\/p>\n

\\(dy\\over dx\\) = -\\(2(ax + hy + g)\\over 2(hx + by + f)\\)<\/p>\n

\\(dy\\over dx\\) = -\\((ax + hy + g)\\over (hx + by + f)\\)<\/p>\n\n\n

\n
Next – Logarithmic Differentiation \u2013 Examples & Formula<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Differentiation of Inverse Trigonometric Functions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the differentiation of implicit function with examples. Let’s begin – Differentiation of Implicit Function If the variables x and y are connected by a relation of the form f(x,y) = 0 and it is not possible or convenient to express y as a function x in the form y\u00a0 …<\/p>\n

Differentiation of Implicit Function<\/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":[36],"tags":[275,321,322],"yoast_head":"\nDifferentiation of Implicit Function - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the differentiation of implicit function 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\/differentiation-of-implicit-function\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differentiation of Implicit Function - 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