Higher Order Partial Derivatives

Since a partial derivative of a function is itself a function, we can take derivatives of it as well. Since it will be a function of more than one variable (usually) we can take partial derivatives of the derivative functions with respect to either variable. Thus, we can compute

$$\frac{\partial }{\partial x}\left(\frac{\partial f}{\partial... ...partial }{\partial y}\left(\frac{\partial f}{\partial y}\right)$$

for a function of x and y. These are all of the possibilities. Usually, they are denoted, respectively, as

$$\frac{\partial^2 f}{\partial x^2} \qquad \frac{\partial^2 f}... ...\partial x \partial y} \qquad \frac{\partial^2 f}{\partial y^2}$$

or as f_xx, f_yx, f_xy and f_yy. For most reasonable functions (ie. those with continuous partial derivatives) the mixed partial derivatives f_xy and f_yx are equal.

For practice, let's compute all of the first and second partials of the function $g(s,t) = 2t\sin s + e^{st}$.

$$g_s = \frac{\partial }{\partial s}(2t\sin s + e^{st}) = 2t \... ...l s} + \frac{\partial e^{st}}{\partial s} = 2t\cos s + t e^{st}$$

$$g_t = \frac{\partial }{\partial t}(2t\sin s + e^{st}) = 2\si... ...al t} + \frac{\partial e^{st}}{\partial t} = 2\sin s + s e^{st}$$

$$g_{tt} = \frac{\partial g_t}{\partial t} = \frac{\partial }{\partial t}(2\sin s + s e^{st}) = s^2e^{st}$$

$$g_{ss} = \frac{\partial g_s}{\partial s} = \frac{\partial }{\partial s}(2t\cos s + t e^{st}) = -2t\sin s + t^2 e^{st}$$

$$g_{st} = \frac{\partial g_s}{\partial t} = \frac{\partial }{\partial t}(2t\cos s + t e^{st}) = 2\cos s + e^{st} + st e^{st}$$

$$g_{ts} = \frac{\partial g_t}{\partial s} = \frac{\partial }{\partial s}(2t\cos s + t e^{st}) = 2\cos s + e^{st} + st e^{st}$$

We can compute third order partial derivatives of f(x,y). There are eight possibilities: f_xxx, f_xxy, f_xyx, f_yxx, f_xyy, f_yxy, f_yyx, and f_yyy. However, for most reasonable functions (those with continuous second partials) f_xxy = f_xyx = f_yxx and f_xyy = f_yxy = f_yyx.

Higher and higher derivatives can be computed for functions of any number of variables, provided that the function is smooth enough. As another quick example, let's list all the first and second partial derivatives of a function of three variables h(x,y,z):

$$\mbox{First order partials:} \qquad h_x, h_y, h_z$$

$$\mbox{Second order partials:} \qquad h_{xx}, h_{yy}, h_{zz}, h_{xy}, h_{xz}, h_{yz}.$$

Note that again, the mixed partials are equal.

Note that we can always return to the good old limit definition of a derivative. To compute $\frac{\partial f(x,y)}{\partial x}$ at the point (x₀, y₀) we have

$$\left. \frac{\partial f}{\partial x} \right\vert _{(x_0,y_0)... ...tarrow 0} \frac{f(x_0 + \Delta x, y_0) - f(x_0,y_0)}{\Delta x}.$$

For obvious reasons, we won't be using this very much. But keep in mind, it's actually the basis for everything that we do with derivatives.