{"id":2490,"date":"2021-07-12T21:03:57","date_gmt":"2021-07-12T15:33:57","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=2490"},"modified":"2026-01-12T16:11:32","modified_gmt":"2026-01-12T10:41:32","slug":"what-is-partial-differentiation","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-partial-differentiation\/","title":{"rendered":"What is partial differentiation?"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Overview<\/strong><\/h2>\n\n\n\n<p>We use partial differentiation in vector calculus and differential geometry.<\/p>\n\n\n\n<p>When the function depends on two or more variables then the derivative changes into partial derivative.<\/p>\n\n\n\n<p>A partial derivative of a function of several variables is an ordinary derivative with respect to one of the variables when all the other variables remain constant.<\/p>\n\n\n\n<p>We use all the rules of differentiation but the condition is that if we are differentiating partially with respect to one variable, all the other variables are treated as constants.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Definition<\/strong> of Partial Differentiation<\/h2>\n\n\n\n<p>Suppose we have a function f(x, y), which is dependent on two variables x and y. x and y are independent of each other.<\/p>\n\n\n\n<p>Then we will say that the function \u2018f\u2019 partially depends on these two variables.<\/p>\n\n\n\n<p>The derivative of the function \u2018f\u2019 is called the partial derivative of \u2018f\u2019.<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"How to do PARTIAL DIFFERENTIATION (Hindi)\" width=\"1170\" height=\"878\" src=\"https:\/\/www.youtube.com\/embed\/m2SogX3AtjI?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">&nbsp;<strong>Notations<\/strong><\/h2>\n\n\n\n<p>Suppose we have a function u = f (x, y) of two independent variables x and y. If we keep y constant and x varies then function u becomes a function of x only. The derivative of u with respect to x, keeping y as constant is called partial derivative of \u2018u\u2019, with respect to \u2018x\u2019.<\/p>\n\n\n\n<p>The following notations we use for partial differentiation-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/Ba3dMXIf8k9-J41gWNAYZczKUnOYlmdsFWZGfCFhtdKKzDzy7njvbC43GmEjA5ke55OKXJCw3VVhZMfPcHYsYdv1hAR3WD89xOSy6n_8VGXAaddn5zN6RB437B0NnN5u8ohgRn80\" alt=\"\"\/><\/figure><\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/AYqVbiOk1NBxCIWgsxx4zTySKUMntC2xGuCjNC1t30qYwYZQ-FO8qvL61vFDTXFnAiMe82rz-lQjHKK6GEGKc9sFvDHI8RHsoBKCdfkj4M2Tu4FGkJ4RYYdOmijBEIDlGtjQNVSv\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Similarly, the partial derivative of \u2018<em>u<\/em>\u2019 with respect to \u2018<em>y<\/em>\u2019 keeping <em>x <\/em>as constant is denoted by<\/p>\n\n\n\n<p class=\"has-text-align-center\"><img loading=\"lazy\" decoding=\"async\" width=\"244\" height=\"124\" src=\"https:\/\/lh5.googleusercontent.com\/wGOGSK3kZSJhnsFEOP5vwsGoocTW7MDY06KIewbQljgyY1XnJ-Z_xZJ0kXgR2ctIjbAo58PcBpMn7BTzTItbbw66u2V4_hzVMAUIdpSyxcxcO7_u0ed03GfePJOFOV4XpN6E34MD\"><\/p>\n\n\n\n<p><strong>Product and quotient rules of partial derivative<\/strong><\/p>\n\n\n\n<p><strong>Product rule:<\/strong><\/p>\n\n\n\n<p>If u = f(x, y).g(x, y), then&nbsp;<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/F77_ULS7gKs-j59NSid95bxFOa9Ol9XsiXEyHXP4OcetbhTFLeUEpOO5Zsi6PXZ8Z2AfVIqgWT4wj93p6jIYyGghDoyZeVRwA5ithb6EiKs83cAN-K7zzZSn_N3-cFHVA3UIsQmy\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>And<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/9SKhQDmL7BRjwaOl5yj3KrBZmHRqFu04V8FiLqaGvjfAkkkxUCHcIzLqU6M_J9z8OAyg7kM9CkU6shG7u3hJWwq-wnauf0tCDcOvf2J4sQYvfMHY60dm71f2qxbStn-kcn1b4np-\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Quotient rule:<\/strong><\/p>\n\n\n\n<p>If u = f(x, y)\/g(x, y), then [note- g(x, y) should be non-zero]<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/POGM4wl9VcFkjkCuuVVT7VucPL4dF-DHc7oqueKK7KDzA5BKsbPHETiClSTRR5B9M6z6I9BNSNv44BvZmKlsKgV1gwspt-csFGpZUT0HKdxWHiSF6gspiUL35DyZuxvkPBntSXYo\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>And<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/m5lR9AwljPwtpUtfvXZVZt51BLjP2aK8Q-4KXGLKvNGu6EfYuoDPXWXAKDU5Fo7M1RxgIuAWzPKs9Vgjx8qtZyNMVFTdW4X_ZNats6NGGVpH-aUb-LBJLDbvcdNuHRvpa58uW6Dj\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Partial derivative of higher orders<\/strong><\/p>\n\n\n\n<p>Let u = f(x, y) then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/irnwKa2Cjk4BsVzcR4BzcHqUfWUNNgPaPf9fpc_j9QaXMSeBchlII4v5mZ7RDr1F8Ka1x0jl4dechn06pu9ab3TMFtWBvcmOHTnEy_P2oq4oML6kghwcscWxTxiUgmhjBVyUb2sT\" width=\"21\" height=\"33\">and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/CHUyopRjSfPq6sKhNfmTwQqJ7vrAxQe3C7BMeK5wHAw430ky0Q-5V01puBq4GlLRBFbe1D5nzYoLodTSW5Mdz9bIHTiL_rzAFB1oAXZeki6L0jOgqqgpG8v-fl7zmB-SshH9VhIj\" width=\"22\" height=\"34\"> will be the function of x and y, and these functions can again be differentiated partially with respect to x and y respectively.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/5LJvRhpoU5BK-f1Y4vfW_FW2zc04OKTQ7w6UaAGHEUVoiP44UpuAyzT8qHYaVjjvFLLLjwINKodLlJbDvh8vMYeoGdu2SpC1sqxBGREQk8ZpS28wMss4bgrO8d2Ns88Ekx7GkWXf\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>These are called the second-order partial derivatives.<\/p>\n\n\n\n<p><strong>Note-&nbsp;<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/4e4hjPOJPwM8JV9HKDGv9e6Dff98Zck5uQpvxwwYKM1hAoTID6wvPuz2Uqtrr9UFfaCEUfkt8UPoqptGtW31Cv8YU_iGPclLa4p4hv_RRDi62ftAlqdv4onibPCOT7T2swkcY3g1\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Solved examples<\/strong><\/p>\n\n\n\n<p><strong>Example: Differentiate the following function partially with respect to x and y.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/t-kHlrtp0qQ6cWsialHVsCly66vKFgx_fkV5mfzFqtGFaGAvz5vfPu5AS2is61IXIU7FQoGINsPkVDn9h2S4TnDLnRwCtL1LsS--6nAqnZ3G4aUZDha06s5FBdEoiQmMmMyl0yWm\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<p>Here the function is-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/t0MmQ0aoMvLNkxRuzQWgSTIVCBwG8hdr4cP-nItlAaESYhnindGpJOAr8wSlTd7ZuibrleAQv0BZ0ogFC86Bz0rsQJ4cr4eaKMK_RyczrBS6g0KwEb-rMgfSjK89Pqk3Pfr4aMUO\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Now differentiating partially with respect to \u2018x\u2019, we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/Uk2XSCn_XzxHrFW_5KxPPahyItDMpfaQ4o2QvnBsLfI1UuvPmRrKomeQH6pDF7t0Dpbg-HNLnHSMDZbE8ExsUWxfY6g1QNIDz-ZcYTqZ3vEghM6ALOhebYdeyNiFNXgcwe6A03bQ\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Similarly differentiating partially with respect to \u2018y\u2019, we get-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/W88fzjjsm94CzaneRo3hMqJm1liHBfKOuwoUU3Mm1gEeBI-7XcJeoj5NyKAiKuFpwTEG3xKOWa5rhwZcknttL3T1Zz4t4hRsKmgJNM64UdNrjA25tBFoxwP6FkfeTtjdVTG3E9ot\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Example: Find the value of <\/strong><strong><\/strong><strong> of the following function-<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh4.googleusercontent.com\/TrgN6Nqp4X_Tkc2tKjGSiJhdhwHsAbffntCA01ee3d3-JUEs5O_fKFIG1Fl8zW6St0pQj1UuzdqmqXXpKluhU8tBLITnRES4ZHI8WoiJMdDpVcicl4qRSzgRcommmXwq1RXolEGX\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Solution:<\/p>\n\n\n\n<p>Here we have-<\/p>\n\n\n\n<figure class=\"wp-block-image is-resized\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/e0WTTtnX4mxrEF_XPyIdVqupp9OPK8HJS1mvU8DbTwhTdXJ1ncL210-Me4rBY5eEINsoD6SxLMnRPZdudiqL_Vc8t9Uk5uPaPh6sU8xbVFltrtzK3g2DemKULMV7wRvRBS_8_Yo-\" alt=\"\" style=\"width:69px;height:23px\"\/><\/figure>\n\n\n\n<p>First we will partially differentiate the given function with respect to \u2018z\u2019<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh3.googleusercontent.com\/JUarZPCtkDUF4UD41UYEeELDC34b-bHVhrEjdlkU-aIWq9azWlpjcdQwM96E5IZJXMiOB3SrpHYHw0d3fBH4eTAw78cdhlkq_cpN2jJPgcHe1dEj3wAG4yOwSjW-G9WqBmQc6-43\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Now<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/-dNM6fAPfDx_u8TO8DbrlNZG-BZnBx_FWQUHB-O7OqDsNZrDaC6r5od-dXUGK_3efcSYCJSZVxOxPUuY3rWcZxgVCVrLzvcBsE0AgfbR_ZC2iS1YF0Bw3eaxnefga1dHgzsJszQe\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Finally<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh5.googleusercontent.com\/7I7CwKu6UNzBXrsGO5iRaoxXjLPsiIYmeFRE6atdQVUQlNFN8avm5EOadEseRE-EOMLjKsdEVa1jVnNQS3huDx_zphxuT_JlNYC2iYy0G_fQA3MEbk6HIvCGUWWU8ilEwPGh7qF0\" alt=\"\"\/><\/figure><\/div>\n\n\n<p>Which can be written as-<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/lh6.googleusercontent.com\/ThgE4Hh253P3cmsXCuAV9wFo1DOgsXZSq7g7oSiguJtDgKW2Jto2tKojV-3lLmNSc4-PdNDdcNnNz8h8h_bfrc8wCiJu3uL9hPqxb9sp2NoBS4Qbia9aTRMQ4JXkngf1Qo_Cj7qN\" alt=\"\"\/><\/figure><\/div>\n\n\n<p><strong>Interested in learning about similar topics? Here are a few hand-picked blogs for you!<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/www.goseeko.com\/blog\/what-is-the-testing-of-hypothesis\">What is Hypothesis?<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/www.meridianpint.com\/touchless-menu\/\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/www.meridianpint.com\/touchless-menu\/<\/a><\/li>\n<\/ul>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>A partial derivative of a function of several variables is an ordinary derivative with respect to one of the variables when all the other variables remain constant.<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[31],"tags":[],"class_list":["post-2490","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v25.3.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is partial differentiation? - Goseeko blog<\/title>\n<meta name=\"description\" content=\"Partial differentiation is used in vector calculus and differential geometry. 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