Calculus III:
Partial Derivatives
Derivatives of functions of more than one variable
Higher Order Partial Derivatives

1. Defnitions and rules

As we had higher order ordinary derivatives with functions of one variable, we will also have higher order partial derivatives of functions of more than one variable. We will consider the case of a function of two variables f(x,y).

The first order partial derivatives with respect to x and with respect to y, of the function f(x,y) are in turn differentiated each with respect to x or y. Here for the case of a function of two variables there will be a total of four possible second order derivatives. Here they are:

2. Example

Find all the four second order derivatives for f(x,y) = cos(x) + xy³

We have:

f_xx = - cos(x) , f_yy = 6xy , f_yx = 3y² , and f_xy = 3y² .

Let’s notice that f_xy = f_yx . This is not always the case.
f_xy and f_yx are equals following the Clairaut’s Theorem:

Let f(x,y) a function defined on a disk D that contains the point (a, b). If the functions f_xy and f_yx are continuous on this disk then f_xy(a,b) = f_yx(a,b)

Alexis Claude Clairaut (1713 – 1765) French mathematician.

3. Hier order derivatives

Here are some third order partial derivatives of function of two variables:

f_xxx, f_xyy , f_xxy, f_xyx , f_yxx ...

For these we differentiate once with respect to y and twice with respect to x, the extension to Clairaut’s Theorem says these functions are equals. That is

f_xxy = f_xyx = f_yxx

We can extend Clairaut’s theorem to any function and mixed partial derivatives. The only requirement is that in each derivative we differentiate with respect to each variable the same number of times. For example: f_xxxxtyy = f_tyyxxxx

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