4.3 Partial derivatives

Now that we have examined limits and continuity of functions of two variables, we can proceed to study derivatives. Finding derivatives of functions of two variables is the key concept in this chapter, with as many applications in mathematics, science, and engineering as differentiation of single-variable functions. However, we have already seen that limits and continuity of multivariable functions have new issues and require new terminology and ideas to deal with them. This carries over into differentiation as well.

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 $y$ as a function of $x.$ Leibniz notation for the derivative is $dy\text{/}dx,$ which implies that $y$ is the dependent variable and $x$ is the independent variable. For a function $z=f(x,y)$ of two variables, $x$ and $y$ are the independent variables and $z$ is the dependent variable. This raises two questions right away: How do we adapt Leibniz 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 $d$ in the original notation is replaced with the symbol $\partial .$ (This rounded $\text{“d”}$ is usually called “partial,” so $\partial f\text{/}\partial x$ is spoken as the “partial of $f$ with respect to $x\text{.”})$ 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.