Unit 1: Partial Derivatives

The Concept

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.

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:

[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]

[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]

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.

The Plot

Now, you should engage with the 3D plot below for partial derivatives^[1]. Follow the steps below to apply changes to the plot and observe the effects:

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]-plane.
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.
3. Repeat the same steps in (1) and (2) for [latex]\frac{\partial z}{\partial y}[/latex].