{"id":5,"date":"2021-12-11T11:32:52","date_gmt":"2021-12-11T16:32:52","guid":{"rendered":"https:\/\/pressbooks.library.ryerson.ca\/multivariatecalculus\/?p=5"},"modified":"2022-03-06T01:48:38","modified_gmt":"2022-03-06T06:48:38","slug":"chapter-1","status":"publish","type":"chapter","link":"https:\/\/pressbooks.library.torontomu.ca\/multivariatecalculus\/chapter\/chapter-1\/","title":{"raw":"Unit 1: Partial Derivatives","rendered":"Unit 1: Partial Derivatives"},"content":{"raw":"

The Concept<\/h2>\r\nWhen studying derivatives of functions of one variable [latex]y=f(x)[\/latex], we found that one interpretation of the derivative is an instantaneous rate of change of [latex]y[\/latex] as a function of [latex]x[\/latex]. Leibniz notation for the derivative is [latex]\\frac{dy}{dx}[\/latex], which implies that [latex]y[\/latex] is the dependent variable and [latex]x[\/latex] is the independent variable. [latex]\\frac{dy}{dx}[\/latex] also represents the slope of the tangent line at a certain point of this function.\r\n\r\nFor a function [latex]z=f(x,y)[\/latex] of two variables, [latex]x[\/latex] and [latex]y[\/latex] are the independent variables (input to function [latex]f[\/latex]) and [latex]z[\/latex] is the dependent variable (output of function [latex]z[\/latex], the value of [latex]z[\/latex] is depend on the values of [latex]x[\/latex] and [latex]y[\/latex] ). We will have two partial derivatives and their Leibniz notations are [latex]\\frac{\\partial z}{\\partial x}[\/latex], and [latex]\\frac{\\partial z}{\\partial y}[\/latex]. They are analogous to ordinary derivatives:\r\n

[latex]\\frac{\\partial z}{\\partial x}(x_0, y_0)=\\frac{\\text{change in } z}{\\text{change in }x}(\\text{holding }y \\text{ as a constant } y_0)[\/latex]<\/p>\r\n

[latex]\\frac{\\partial z}{\\partial y}(x_0, y_0)=\\frac{\\text{change in } z}{\\text{change in }y}(\\text{holding }x \\text{ as a constant } x_0)[\/latex]<\/p>\r\nBesides the Leibniz notations above, you can also write the derivatives as [latex]\\frac{\\partial z}{\\partial x}=f_x[\/latex] and [latex]\\frac{\\partial z}{\\partial y}=f_y[\/latex]. Similar to the geometric meaning of [latex]\\frac{dy}{dx}[\/latex] in two-dimensional (2D), [latex]\\frac{\\partial z}{\\partial x}[\/latex] and [latex]\\frac{\\partial z}{\\partial y}[\/latex] in three-dimensional (3D) represent the slopes of tangent lines as well.\r\n

The Plot<\/h2>\r\nNow, you should engage with the 3D plot below for partial derivatives[footnote]Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.[\/footnote]. Follow the steps below to apply changes to the plot and observe the effects:\r\n
    \r\n \t
  1. Input a function of two variables, then set [latex]y[\/latex] as a constant, e.g., [latex]-1[\/latex]. A cross-section plane [latex]y=-1[\/latex] is plotted. Recall that the function [latex]y = y_0[\/latex] (or [latex]x=x_0[\/latex]) in 3D represents the planes that are perpendicular to the [latex]xy[\/latex]<\/em>-plane.<\/li>\r\n \t
  2. A tangent line passing through the point ([latex]x_0[\/latex], [latex]y_0[\/latex]) and also on the cross-section plane [latex]y=y_0[\/latex] is plotted. Change the [latex]y[\/latex]-values using the slider, and you will see the cross-section and the tangent line changes. You can also rotate the graph to get a better view. Since the particle derivative is the slope of the tangent line, the partial derivative [latex]\\frac{\\partial z}{\\partial x}[\/latex] changes as well.<\/li>\r\n \t
  3. Repeat the same steps in (1) and (2) for [latex]\\frac{\\partial z}{\\partial y}[\/latex].<\/li>\r\n<\/ol>\r\n[h5p id=\"2\"]<\/span>\r\n
    \r\n

    Self-Checking Questions<\/h2>\r\n<\/header>\r\n
    \r\n\r\nCheck your understanding by solving the following questions[footnote]Gilbert Strang, Edwin \u201cJed\u201d Herman,\u00a0 OpenStax,\u00a0Calculus Volume 3,\u00a0 Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).[\/footnote]<\/span>:\r\n
      \r\n \t
    1. Let [latex]f(x,y)=\\frac{xy}{x-y}[\/latex]. Find [latex]f_x(2,-2)[\/latex] and [latex]f_y(2,-2)[\/latex].<\/li>\r\n \t
    2. The apparent temperature index, [latex]A[\/latex], is a measure of how the temperature feels,<\/li>\r\n<\/ol>\r\n

      [latex]A=0.885x -22.4 y +1.2 xy -0.544[\/latex]<\/p>\r\n

      where [latex]x[\/latex] is relative humidity and [latex]y[\/latex] is the air temperature. Find [latex]\\frac{\\partial A}{\\partial x}[\/latex] and [latex]\\frac{\\partial A}{\\partial y}[\/latex] when [latex]x=20\u00b0F[\/latex] and [latex]y=1[\/latex].<\/p>\r\n[h5p id=\"7\"]<\/span><\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n ","rendered":"

      The Concept<\/h2>\n

      When studying derivatives of functions of one variable [latex]y=f(x)[\/latex], we found that one interpretation of the derivative is an instantaneous rate of change of [latex]y[\/latex] as a function of [latex]x[\/latex]. Leibniz notation for the derivative is [latex]\\frac{dy}{dx}[\/latex], which implies that [latex]y[\/latex] is the dependent variable and [latex]x[\/latex] is the independent variable. [latex]\\frac{dy}{dx}[\/latex] also represents the slope of the tangent line at a certain point of this function.<\/p>\n

      For a function [latex]z=f(x,y)[\/latex] of two variables, [latex]x[\/latex] and [latex]y[\/latex] are the independent variables (input to function [latex]f[\/latex]) and [latex]z[\/latex] is the dependent variable (output of function [latex]z[\/latex], the value of [latex]z[\/latex] is depend on the values of [latex]x[\/latex] and [latex]y[\/latex] ). We will have two partial derivatives and their Leibniz notations are [latex]\\frac{\\partial z}{\\partial x}[\/latex], and [latex]\\frac{\\partial z}{\\partial y}[\/latex]. They are analogous to ordinary derivatives:<\/p>\n

      [latex]\\frac{\\partial z}{\\partial x}(x_0, y_0)=\\frac{\\text{change in } z}{\\text{change in }x}(\\text{holding }y \\text{ as a constant } y_0)[\/latex]<\/p>\n

      [latex]\\frac{\\partial z}{\\partial y}(x_0, y_0)=\\frac{\\text{change in } z}{\\text{change in }y}(\\text{holding }x \\text{ as a constant } x_0)[\/latex]<\/p>\n

      Besides the Leibniz notations above, you can also write the derivatives as [latex]\\frac{\\partial z}{\\partial x}=f_x[\/latex] and [latex]\\frac{\\partial z}{\\partial y}=f_y[\/latex]. Similar to the geometric meaning of [latex]\\frac{dy}{dx}[\/latex] in two-dimensional (2D), [latex]\\frac{\\partial z}{\\partial x}[\/latex] and [latex]\\frac{\\partial z}{\\partial y}[\/latex] in three-dimensional (3D) represent the slopes of tangent lines as well.<\/p>\n

      The Plot<\/h2>\n

      Now, you should engage with the 3D plot below for partial derivatives[1]<\/sup><\/a>. Follow the steps below to apply changes to the plot and observe the effects:<\/p>\n

        \n
      1. Input a function of two variables, then set [latex]y[\/latex] as a constant, e.g., [latex]-1[\/latex]. A cross-section plane [latex]y=-1[\/latex] is plotted. Recall that the function [latex]y = y_0[\/latex] (or [latex]x=x_0[\/latex]) in 3D represents the planes that are perpendicular to the [latex]xy[\/latex]<\/em>-plane.<\/li>\n
      2. A tangent line passing through the point ([latex]x_0[\/latex], [latex]y_0[\/latex]) and also on the cross-section plane [latex]y=y_0[\/latex] is plotted. Change the [latex]y[\/latex]-values using the slider, and you will see the cross-section and the tangent line changes. You can also rotate the graph to get a better view. Since the particle derivative is the slope of the tangent line, the partial derivative [latex]\\frac{\\partial z}{\\partial x}[\/latex] changes as well.<\/li>\n
      3. Repeat the same steps in (1) and (2) for [latex]\\frac{\\partial z}{\\partial y}[\/latex].<\/li>\n<\/ol>\n

        <\/p>\n

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

        <\/span><\/p>\n

        \n
        \n

        Self-Checking Questions<\/h2>\n<\/header>\n
        \n

        Check your understanding by solving the following questions[2]<\/sup><\/a><\/span>:<\/p>\n

          \n
        1. Let [latex]f(x,y)=\\frac{xy}{x-y}[\/latex]. Find [latex]f_x(2,-2)[\/latex] and [latex]f_y(2,-2)[\/latex].<\/li>\n
        2. The apparent temperature index, [latex]A[\/latex], is a measure of how the temperature feels,<\/li>\n<\/ol>\n

          [latex]A=0.885x -22.4 y +1.2 xy -0.544[\/latex]<\/p>\n

          where [latex]x[\/latex] is relative humidity and [latex]y[\/latex] is the air temperature. Find [latex]\\frac{\\partial A}{\\partial x}[\/latex] and [latex]\\frac{\\partial A}{\\partial y}[\/latex] when [latex]x=20\u00b0F[\/latex] and [latex]y=1[\/latex].<\/p>\n

          <\/p>\n

          \n