Partial Derivatives | Calculus III (2024)

Derivatives of a Function of Two Variables

When studying derivatives of functions of one variable, 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]dy/dx[/latex], which implies that [latex]y[/latex] is the dependent variable and [latex]x[/latex] is the independent variable. For a function [latex]z=f\,(x,\ y)[/latex] of two variables, [latex]x[/latex] and [latex]y[/latex] are the independent variables and [latex]z[/latex] is the dependent variable. This raises two questions right away: How do we adaptLeibniz notation for functions of two variables? Also, what is an interpretation of the derivative? The answer lies in partial derivatives.

This definition shows two differences already. First, the notation changes, in the sense that we still use a version of Leibniz notation, but the [latex]d[/latex] in the original notation is replaced with the symbol [latex]\partial[/latex]. (This rounded “[latex]d[/latex]” is usually called “partial,” so [latex]\partial f/\partial x[/latex] is spoken as the “partial of [latex]f[/latex] with respect to [latex]x[/latex].”)This is the first hint that we are dealing with partial derivatives. Second, we now have two different derivatives we can take, since there are two different independent variables. Depending on which variable we choose, we can come up with different partial derivatives altogether, and often do.

The idea to keep in mind when calculating partial derivatives is to treat all independent variables, other than the variable with respect to which we are differentiating, as constants. Then proceed to differentiate as with a function of a single variable. To see why this is true, first fix [latex]y[/latex] and define [latex]g\,(x)=f\,(x,\ y)[/latex] as a function of [latex]x[/latex]. Then

[latex]g'\,(x)=\displaystyle\lim_{h\to 0}\frac{g\,(x+h)-g\,(x)}{h}=\displaystyle\lim_{h\to 0}\frac{f\,(x+h,\ y)-f\,(x,\ y)}{h}=\frac{\partial f}{\partial x}.[/latex]

The same is true for calculating the partial derivative of [latex]f[/latex] with respect to [latex]y[/latex]. This time, fix [latex]x[/latex] and define [latex]h\,(y)=f\,(x,\ y)[/latex] as a function of [latex]y.[/latex] Then

[latex]h'\,(x)=\displaystyle\lim_{k\to 0}\frac{h\,(x+h)-h\,(x)}{k}=\displaystyle\lim_{k\to 0}\frac{f\,(x,\ y+k)-f\,(x,\ y)}{k}=\frac{\partial f}{\partial y}.[/latex]

All differentiation rules from the Introduction to Derivatives apply.

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How can we interpret these partial derivatives? Recall that the graph of a function of two variables is a surface in [latex]\mathbb{R}^{3}[/latex]. If we remove the limit from the definition of the partial derivative with respect to [latex]x[/latex], the difference quotient remains:

[latex]\LARGE{\frac{f\,(x+h,\ y)-f\,(x,\ y)}{h}}[/latex].

This resembles the difference quotient for the derivative of a function of one variable, except for the presence of the [latex]y[/latex] variable.variable. Figure 1 illustrates a surface described by an arbitrary function [latex]z=f\,(x,\ y)[/latex].

In Figure 1, the value of [latex]h[/latex] is positive. If we graph [latex]f\,(x,\ y)[/latex] and [latex]f\,(x+h,\ y)[/latex] for an arbitrary point [latex](x,\ y)[/latex], then the slope of the secant line passing through these two points is given by

[latex]\LARGE{\frac{f\,(x+h,\ y)-f\,(x,\ y)}{h}}[/latex].

This line is parallel to the [latex]x[/latex]-axis. Therefore, the slope of the secant line represents an average rate of change of the function [latex]f[/latex] as we travel parallel to the [latex]x[/latex]-axis. As [latex]h[/latex] approaches zero, the slope of the secant line approaches the slope of the tangent line.

If we chose to change [latex]y[/latex] instead of [latex]x[/latex] by the same incremental value [latex]h[/latex], then the secant line is parallel to the [latex]y[/latex]-axisw and so is the tangent line. Therefore, [latex]\partial f/\partial x[/latex] represents the slope of the tangent line passing through the point [latex](x,\ y,\ f\,(x,\ y))[/latex] parallel to the [latex]x[/latex]-axis and [latex]\partial f/\partial y[/latex] represents the slope of the tangent line passing through the point[latex](x,\ y,\ f\,(x,\ y))[/latex] parallel to the [latex]y[/latex]-axis. If we wish to find the slope of a tangent line passing through the same point in any other direction, then we need what are calleddirectional derivatives, which we discuss in Directional Derivatives and the Gradient.

We now return to the idea of contour maps, which we introduced in Functions of Several Variables. We can use a contour map to estimate partial derivatives of a function [latex]g\,(x,\ y)[/latex].

Functions of More Than Two Variables

Suppose we have a function of three variables, such as [latex]w=f\,(x,\ y,\ z)[/latex]. We can calculate partial derivatives of [latex]w[/latex] with respect to any of the independent variables, simply as extensions of the definitions for partial derivatives of functions of two variables.

We can calculate a partial derivative of a function of three variables using the same idea we used for a function of two variables. Fore example, if we have a function [latex]f[/latex] of [latex]x,\ y[/latex], and [latex]z[/latex], and we wish to calculate [latex]\partial f/\partial x[/latex], then we create the other two independent variables as if they are constants, then differentiate with respect to [latex]x[/latex].

Watch the following video to see the worked solution to the above Try It