{"id":666082,"date":"2023-08-02T11:17:00","date_gmt":"2023-08-02T05:47:00","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=666082"},"modified":"2023-08-02T11:41:52","modified_gmt":"2023-08-02T06:11:52","slug":"partial-derivative-3","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/","title":{"rendered":"Partial derivative"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Introduction_to_Partial_Derivative\" title=\"Introduction to Partial Derivative\">Introduction to Partial Derivative<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Definition_of_Partial_derivative\" title=\"Definition of Partial derivative\">Definition of Partial derivative<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Symbol_for_partial_derivative\" title=\"Symbol for partial derivative\">Symbol for partial derivative<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Partial_derivative_formula\" title=\"Partial derivative formula\">Partial derivative formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Partial_differentiation_with_example\" title=\"Partial differentiation with example\">Partial differentiation with example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Rules_in_partial_differentiation\" title=\"Rules in partial differentiation\">Rules in partial differentiation<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Frequently_asked_questions_on_Partial_differentiation\" title=\"Frequently asked questions on Partial differentiation\">Frequently asked questions on Partial differentiation<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#What_is_partial_derivative_with_example\" title=\"What is partial derivative with example? \">What is partial derivative with example? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#What_is_partial_derivative_formula\" title=\"What is partial derivative formula? \">What is partial derivative formula? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#What_does_%E2%88%82_means_in_math\" title=\"What does \u2202 means in math? \">What does \u2202 means in math? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#What_is_the_difference_between_partial_derivative_and_ordinary_derivative\" title=\"What is the difference between partial derivative and ordinary derivative? \">What is the difference between partial derivative and ordinary derivative? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#How_many_types_of_partial_derivatives_are_there\" title=\"How many types of partial derivatives are there? \">How many types of partial derivatives are there? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Who_invented_partial_derivative\" title=\"Who invented partial derivative? \">Who invented partial derivative? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Why_do_we_need_partial_differential_equation\" title=\"Why do we need partial differential equation? \">Why do we need partial differential equation? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#What_are_the_methods_of_solving_partial_differential_equations\" title=\"What are the methods of solving partial differential equations? \">What are the methods of solving partial differential equations? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/#Is_d_and_%E2%88%82_are_same\" title=\"Is d and \u2202 are same? \">Is d and \u2202 are same? <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Partial_Derivative\"><\/span>Introduction to Partial Derivative<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In multivariable calculus, partial derivatives are a key idea. They calculate the rate of change of a function with respect to one variable while maintaining the others constant. They are denoted by and allow us to see how a function responds to individual inputs. Partial derivatives are critical in understanding complicated systems and optimising solutions in domains such as physics and engineering.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_Partial_derivative\"><\/span>Definition of Partial derivative<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A partial derivative is a multivariable calculus term that assesses the rate of change of a function with respect to one variable while holding all other variables constant. It aids in analysing how a function responds to specific inputs and is used in many domains, including physics and engineering, to better understand complicated systems and optimise solutions..<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Symbol_for_partial_derivative\"><\/span>Symbol for partial derivative<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The symbol used to represent a partial derivative is \u2202 (the partial derivative symbol). It is written before the function with respect to which the partial derivative is being taken and is followed by the variable with respect to which the differentiation is performed. For example, the partial derivative of a function f(x, y) with respect to x is denoted as \u2202f\/\u2202x.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Partial_derivative_formula\"><\/span>Partial derivative formula<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If the function <em>f (x, y)<\/em> is a two variable function in terms of <em>x, y<\/em> then the partial derivative of <em>f<\/em> with respect to <em>x<\/em> can be denoted by <em>fx<\/em> and is defined as<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-666090 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula.png\" alt=\"Partial derivative formula\" width=\"397\" height=\"84\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula.png 397w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula-300x63.png 300w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/p>\n<p>If the function <em>f (x, y)<\/em> is a two variable function in terms of <em>x, y<\/em> then the partial derivative of <em>f<\/em> with respect to <em>y<\/em> can be denoted by <em>fx<\/em> and is defined as<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-666091 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula-1.png\" alt=\"Partial derivative formula\" width=\"397\" height=\"84\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula-1.png 397w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Partial-derivative-formula-1-300x63.png 300w\" sizes=\"(max-width: 397px) 100vw, 397px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/articles\/math-articles\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Articles<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/formulas\/math-formulas\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Formulas<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/logarithm\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Logarithm<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/locus\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Locus<\/button><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Partial_differentiation_with_example\"><\/span>Partial differentiation with example<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Partial differentiation involves finding the partial derivatives of a function with respect to one of its variables while keeping other variables constant.<\/p>\n<p><strong>Example 1<\/strong><\/p>\n<p>Consider the function f(x, y) = x^2 + 3xy + y^2<\/p>\n<p>To find the partial derivatives of this function with respect to x and y, we perform the following differentiations:<\/p>\n<p>Partial derivative with respect to x (keeping y constant):<\/p>\n<p>\u2202f\/\u2202x = d\/dx (x^2 + 3xy + y^2) = 2x + 3y.<\/p>\n<p>Partial derivative with respect to y (keeping x constant):<\/p>\n<p>\u2202f\/\u2202y = d\/dy (x^2 + 3xy + y^2) = 3x + 2y.<\/p>\n<p><strong>Example2<\/strong><\/p>\n<p>Consider the function g(u, v, w) = u^3 + 2u^2v + v^2w.<\/p>\n<p>To find the partial derivatives of this function with respect to u, v, and w, we perform the following differentiations:<\/p>\n<p>Partial derivative with respect to u (keeping v and w constant):<\/p>\n<p>\u2202g\/\u2202u = d\/du (u^3 + 2u^2v + v^2w) = 3u^2 + 4uv.<\/p>\n<p>Partial derivative with respect to v (keeping u and w constant):<\/p>\n<p>\u2202g\/\u2202v = d\/dv (u^3 + 2u^2v + v^2w) = 2u^2 + 2vw.<\/p>\n<p>Partial derivative with respect to w (keeping u and v constant):<\/p>\n<p>\u2202g\/\u2202w = d\/dw (u^3 + 2u^2v + v^2w) = v^2.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Rules_in_partial_differentiation\"><\/span>Rules in partial differentiation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Partial differentiation has some rules that make it easier to compute derivatives of multivariable functions. Here are the key rules for partial differentiation:<\/p>\n<ul>\n<li><strong>Constant Rule:<\/strong> The partial derivative of a constant with respect to any variable is always zero.<br \/>\nFor example, if c is a constant, then<br \/>\n\u2202c\/\u2202x = \u2202c\/\u2202y = \u2202c\/\u2202z = 0.<\/li>\n<li><strong>Power Rule:<\/strong> To find the partial derivative of a variable raised to a constant power with respect to that variable, multiply the constant power by the variable raised to the power decreased by one.<br \/>\nFor example : \u2202\/\u2202x (x^n) = n * x^(n-1) where n is a constant.<\/li>\n<li><strong>Sum\/Difference Rule:<\/strong> The partial derivative of a sum or difference of two functions is the sum or difference of their individual partial derivatives.<br \/>\nFor example: \u2202\/\u2202x (f(x, y) + g(x, y)) = \u2202f\/\u2202x + \u2202g\/\u2202x.<\/li>\n<li><strong>Product Rule:<\/strong>To find the partial derivative of the product of two functions with respect to a variable, use the following formula:<br \/>\nFor example: \u2202\/\u2202x (f(x, y) * g(x, y)) = f(x, y) * \u2202g\/\u2202x + g(x, y) * \u2202f\/\u2202x.<\/li>\n<li><strong>Quotient Rule: <\/strong>To find the partial derivative of the quotient of two functions with respect to a variable, use the following formula:<br \/>\n\u2202\/\u2202x (f(x, y) \/ g(x, y))<br \/>\n= (g(x, y) * \u2202f\/\u2202x &#8211; f(x, y) * \u2202g\/\u2202x) \/ [g(x, y)]^2.<\/li>\n<li><strong>Chain Rule:<\/strong> The chain rule applies to partial differentiation, just as it does in single-variable calculus. If a function is composed of two or more other functions, then the partial derivative of the composite function with respect to a variable is the derivative of the outer function evaluated at the inner function, multiplied by the partial derivative of the inner function with respect to the variable.<br \/>\nFor example:If z = f(u, v) and u = g(x, y), then \u2202z\/\u2202x = (\u2202f\/\u2202u * \u2202g\/\u2202x) + (\u2202f\/\u2202v * \u2202v\/\u2202x).<\/li>\n<li><strong>Mixed Partial Derivatives:<\/strong>If a function has multiple variables, you can find mixed partial derivatives, which are partial derivatives taken successively with respect to different variables. The order in which you take the partial derivatives matters, and sometimes the mixed partial derivatives may not be equal (in general, they are equal if the function has continuous second partial derivatives).<br \/>\nFor example:\u2202^2f\/\u2202x\u2202y represents the partial derivative of f with respect to x and then y.<\/li>\n<\/ul>\n<p>These rules are essential tools in calculating partial derivatives of various functions and are commonly used in applications across mathematics, physics, engineering, economics, and other fields dealing with multivariable functions.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Frequently_asked_questions_on_Partial_differentiation\"><\/span>Frequently asked questions on Partial differentiation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_partial_derivative_with_example\"><\/span>What is partial derivative with example? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\t partial derivative is a derivative of a multivariable function with respect to one of its variables, while holding all other variables constant. In other words, it measures the rate of change of the function concerning one specific variable while treating all other variables as constants. The notation used for partial derivatives involves using the symbol \u2202 (the partial derivative symbol) and subscript to indicate the variable with respect to which the differentiation is performed. Let's go through an example to illustrate partial derivatives: partial derivative is a derivative of a multivariable function with respect to one of its variables, while holding all other variables constant. In other words, it measures the rate of change of the function concerning one specific variable while treating all other variables as constants. The notation used for partial derivatives involves using the symbol \u2202 (the partial derivative symbol) and subscript to indicate the variable with respect to which the differentiation is performed. Let's go through an example to illustrate partial derivatives: Example: Consider the function f(x, y) = x^2 + 3xy + y^2. To find the partial derivatives of this function with respect to x and y, we perform the following differentiations: Partial derivative with respect to x (keeping y constant): \u2202f\/\u2202x = d\/dx (x^2 + 3xy + y^2) = 2x + 3y. Partial derivative with respect to y (keeping x constant): \u2202f\/\u2202y = d\/dy (x^2 + 3xy + y^2) = 3x + 2y. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_partial_derivative_formula\"><\/span>What is partial derivative formula? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe partial derivative formula computes a multivariable function's rate of change with respect to one variable while considering the other variables as constants. It is represented by the symbol f\/xi, where f is the function and xi is the variable of interest. Standard differentiation procedures are applied to the function with regard to xi in the formula. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_does_%E2%88%82_means_in_math\"><\/span>What does \u2202 means in math? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe partial derivative is represented by the symbol \u2202 in mathematics. In multivariable calculus, the partial derivative determines the rate of change of a function with respect to one variable while maintaining all other variables constant. It is utilised when a function is dependent on numerous factors and allows us to see how the function varies in relation to certain variables. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_difference_between_partial_derivative_and_ordinary_derivative\"><\/span>What is the difference between partial derivative and ordinary derivative? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe partial derivative computes how a multivariable function changes in one variable while remaining constant in the others. It's indicated by \u2202\/\u2202x The ordinary derivative, on the other hand, computes the rate of change of a single-variable function with respect to that variable. It's represented by d\/dx. The primary contrast is that partial derivatives deal with functions with several variables, whereas conventional derivatives deal with functions with a single variable. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_many_types_of_partial_derivatives_are_there\"><\/span>How many types of partial derivatives are there? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tPartial derivatives are classified into two types: First-order partial derivatives are a multivariable function's partial derivatives with respect to each of its variables. There are n first-order partial derivatives for a function with n variables. Higher-order partial derivatives: These are created by taking partial derivatives of a function one after the other. For example, the partial derivative of the first-order partial derivative with regard to the same variable is the second-order partial derivative. This procedure can be repeated to get third-order, fourth-order, and higher-order partial derivatives. In summary, partial derivatives are categorised depending on their order, with the two primary categories being first-order partial derivatives and higher-order partial derivatives. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Who_invented_partial_derivative\"><\/span>Who invented partial derivative? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tOver time, numerous mathematicians independently constructed partial derivatives. In the 18th and 19th centuries, Leonhard Euler, Joseph-Louis Lagrange, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann made substantial contributions to the concept's evolution, advancing calculus and mathematical analysis. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Why_do_we_need_partial_differential_equation\"><\/span>Why do we need partial differential equation? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tPartial differential equations (PDEs) are crucial for characterising complicated systems with several interdependent variables. They are used to simulate phenomena like as heat transfer, fluid movement, electromagnetic fields, and quantum mechanics in physics, engineering, finance, and other areas. Solving PDEs aids in understanding and predicting real-world behaviour, allowing for technological and scientific improvements. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_methods_of_solving_partial_differential_equations\"><\/span>What are the methods of solving partial differential equations? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tDepending on the kind of equation and boundary conditions, many methods are used to solve partial differential equations (PDEs). Separation of variables, method of characteristics, Fourier and Laplace transforms, finite difference methods, numerical techniques such as finite element method and finite volume method, and perturbation methods are all common approaches. Each strategy has advantages and disadvantages in certain situations. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_d_and_%E2%88%82_are_same\"><\/span>Is d and \u2202 are same? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIn the context of calculus and differentiation, d and \u2202 represent different types of derivatives. d (d\/dx): The symbol d represents the ordinary derivative with respect to a single variable. When we write d\/dx, it means the derivative of a function with respect to the variable x. \u2202 (\u2202f\/\u2202x): The symbol \u2202 represents the partial derivative. It is used when a function depends on multiple variables. \u2202f\/\u2202x denotes the partial derivative of the function f with respect to the variable x, while holding all other variables constant. So, d and \u2202 are not the same; they represent different types of derivatives: ordinary derivative and partial derivative, respectively. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is partial derivative with example? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"partial derivative is a derivative of a multivariable function with respect to one of its variables, while holding all other variables constant. In other words, it measures the rate of change of the function concerning one specific variable while treating all other variables as constants. The notation used for partial derivatives involves using the symbol \u2202 (the partial derivative symbol) and subscript to indicate the variable with respect to which the differentiation is performed. Let's go through an example to illustrate partial derivatives: partial derivative is a derivative of a multivariable function with respect to one of its variables, while holding all other variables constant. In other words, it measures the rate of change of the function concerning one specific variable while treating all other variables as constants. The notation used for partial derivatives involves using the symbol \u2202 (the partial derivative symbol) and subscript to indicate the variable with respect to which the differentiation is performed. Let's go through an example to illustrate partial derivatives: Example: Consider the function f(x, y) = x^2 + 3xy + y^2. To find the partial derivatives of this function with respect to x and y, we perform the following differentiations: Partial derivative with respect to x (keeping y constant): \u2202f\/\u2202x = d\/dx (x^2 + 3xy + y^2) = 2x + 3y. Partial derivative with respect to y (keeping x constant): \u2202f\/\u2202y = d\/dy (x^2 + 3xy + y^2) = 3x + 2y.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is partial derivative formula? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The partial derivative formula computes a multivariable function's rate of change with respect to one variable while considering the other variables as constants. It is represented by the symbol f\/xi, where f is the function and xi is the variable of interest. Standard differentiation procedures are applied to the function with regard to xi in the formula.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What does \u2202 means in math? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The partial derivative is represented by the symbol \u2202 in mathematics. In multivariable calculus, the partial derivative determines the rate of change of a function with respect to one variable while maintaining all other variables constant. It is utilised when a function is dependent on numerous factors and allows us to see how the function varies in relation to certain variables.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the difference between partial derivative and ordinary derivative? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The partial derivative computes how a multivariable function changes in one variable while remaining constant in the others. It's indicated by \u2202\/\u2202x The ordinary derivative, on the other hand, computes the rate of change of a single-variable function with respect to that variable. It's represented by d\/dx. The primary contrast is that partial derivatives deal with functions with several variables, whereas conventional derivatives deal with functions with a single variable.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How many types of partial derivatives are there? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Partial derivatives are classified into two types: First-order partial derivatives are a multivariable function's partial derivatives with respect to each of its variables. There are n first-order partial derivatives for a function with n variables. Higher-order partial derivatives: These are created by taking partial derivatives of a function one after the other. For example, the partial derivative of the first-order partial derivative with regard to the same variable is the second-order partial derivative. This procedure can be repeated to get third-order, fourth-order, and higher-order partial derivatives. In summary, partial derivatives are categorised depending on their order, with the two primary categories being first-order partial derivatives and higher-order partial derivatives.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Who invented partial derivative? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Over time, numerous mathematicians independently constructed partial derivatives. In the 18th and 19th centuries, Leonhard Euler, Joseph-Louis Lagrange, Augustin-Louis Cauchy, Karl Weierstrass, and Bernhard Riemann made substantial contributions to the concept's evolution, advancing calculus and mathematical analysis.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Why do we need partial differential equation? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Partial differential equations (PDEs) are crucial for characterising complicated systems with several interdependent variables. They are used to simulate phenomena like as heat transfer, fluid movement, electromagnetic fields, and quantum mechanics in physics, engineering, finance, and other areas. Solving PDEs aids in understanding and predicting real-world behaviour, allowing for technological and scientific improvements.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the methods of solving partial differential equations? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Depending on the kind of equation and boundary conditions, many methods are used to solve partial differential equations (PDEs). Separation of variables, method of characteristics, Fourier and Laplace transforms, finite difference methods, numerical techniques such as finite element method and finite volume method, and perturbation methods are all common approaches. Each strategy has advantages and disadvantages in certain situations.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is d and \u2202 are same? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"In the context of calculus and differentiation, d and \u2202 represent different types of derivatives. d (d\/dx): The symbol d represents the ordinary derivative with respect to a single variable. When we write d\/dx, it means the derivative of a function with respect to the variable x. \u2202 (\u2202f\/\u2202x): The symbol \u2202 represents the partial derivative. It is used when a function depends on multiple variables. \u2202f\/\u2202x denotes the partial derivative of the function f with respect to the variable x, while holding all other variables constant. So, d and \u2202 are not the same; they represent different types of derivatives: ordinary derivative and partial derivative, respectively.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Partial Derivative In multivariable calculus, partial derivatives are a key idea. They calculate the rate of change of [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Partial derivative","_yoast_wpseo_title":"Partial Derivative - Definition, Symbol, Formula, Rules and Examples","_yoast_wpseo_metadesc":"Partial derivative measures how a function changes with respect to one variable, treating other variables as constants.","custom_permalink":"articles\/partial-derivative\/"},"categories":[8442,8443],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Partial Derivative - Definition, Symbol, Formula, Rules and Examples<\/title>\n<meta name=\"description\" content=\"Partial derivative measures how a function changes with respect to one variable, treating other variables as constants.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/partial-derivative\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Partial Derivative - 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