Partial Derivatives

For a multivariable function \(f \colon \mathbf{R}^2 \to \mathbf{R}\) its partial derivatives with respect to \(x\) and \(y\) are defined and denoted as:

\(\displaystyle f_x(x,y) = \lim\limits_{h \to 0} \frac{f(x+h, y) \!-\! f(x,y)}{h} \)
\(\displaystyle \biggl( f_x = \operatorname{D}_x(f) = \frac{\partial}{\partial x}f = \frac{\partial f}{\partial x} \biggr) \)
\( \displaystyle f_y(x,y) = \lim\limits_{h \to 0} \frac{f(x, y+h) \!-\! f(x,y)}{h} \)
\( \displaystyle \biggl( f_y = \operatorname{D}_y(f) = \frac{\partial}{\partial y}f = \frac{\partial f}{\partial y} \biggr) \)

These are only “partial” derivatives because the “full derivative” of a multivariable function is not just a single function. These partial derivatives represent the rates-of-change of \(f\) in the \(x\)- and \(y\)-direction or, geometrically, the slopes of the lines tangent to the graph of \(f\) in those directions. To determine formulas for these partial derivatives from a formula for \(f,\) all the rules of single-variable differentiation can be applied while “pretending” the other variable is just a constant.

The second-order partial derivatives of \(f,\) the results of differentiating \(f\) twice with respect to some order of \(x\) and \(y,\) are denoted as: \[ \begin{matrix} \displaystyle f_{xx} = \bigl(f_x\bigr)_x = \operatorname{D}^2_x(f) = \frac{\partial^2}{\partial x^2}f = \frac{\partial^2 f}{\partial x^2} \qquad& \displaystyle f_{xy} = \bigl(f_x\bigr)_y = \operatorname{D}_y\bigl(\operatorname{D}_x(f)\bigr) = \frac{\partial}{\partial y}\frac{\partial}{\partial x}f = \frac{\partial^2 f}{\partial y\partial x} \\[1em] \displaystyle f_{yx} = \bigl(f_y\bigr)_x = \operatorname{D}_x\bigl(\operatorname{D}_y(f)\bigr) = \frac{\partial}{\partial x}\frac{\partial}{\partial y}f = \frac{\partial^2 f}{\partial x\partial y} \qquad& \displaystyle f_{yy} = \bigl(f_y\bigr)_y = \operatorname{D}^2_y(f) = \frac{\partial^2}{\partial y^2}f = \frac{\partial^2 f}{\partial y^2} \end{matrix} \] The partial derivatives \(f_{xx}\) and \(f_{yy}\) can be thought of as representing the concavity of the graph of \(f\) in the \(x\) and \(y\) direction respectively, while the mixed partials \(f_{xy}\) and \(f_{yx}\) represent a “rotational” rate-of-change around a point. And each of these mixed partials represent the same rate-of-change per Clairaut’s Theorem: if \(f\) is defined on some neighborhood containing the point \((a,b)\) and the second-order partial derivatives are all continuous in that neighborhood, then \(f_{xy}(a,b) = f_{yx}(a,b).\)