Higher order partial derivatives

Notes:

2^nd Order Partial Derivatives
Since we can have higher order derivatives on a one variable function, we can also have this for multi-variable functions. We will specifically look at 2^nd order partial derivatives here. For 2^nd order partial derivatives, there are 4 types:

$f_{xx} = \frac{d}{dx}(\frac{df}{dx}) = \frac{d^2f}{dx^2}$
$f_{xy} = \frac{d}{dy}(\frac{df}{dx}) = \frac{d^2f}{dydx}$
$f_{yy} = \frac{d}{dy}(\frac{df}{dy}) = \frac{d^2f}{dy^2}$
$f_{yx} = \frac{d}{dx}(\frac{df}{dy}) = \frac{d^2f}{dxdy}$

Where:
$f_{xx} \to$ derivative in respect to $x$ 2 times
$f_{yy} \to$ derivative in respect to $y$ 2 times
$f_{xy} \to$ derivative in respect to $x$ first, and then respect to $y$
$f_{yx} \to$ derivative in respect to $y$ first, and then respect to $x$

Higher Order Partial Derivatives
Of course, we can have even higher order partial derivatives. For example, we can have:

$f_{xxx} = \frac{d}{dx} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dx^3}$
$f_{xxy} = \frac{d}{dy} (\frac{d^2f}{dx^2}) = \frac{d^3f}{dydx^2}$
$f_{xxxxx} = \frac{d}{dx} (\frac{d^4f}{dx^4}) = \frac{d^5f}{dx^5}$

We cannot list them all here because there is an infinite amount of higher order partial derivatives.

Clairaut's Theorem Suppose that $f$ is 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

$f_{xy}(a,b) = f_{yx}(a,b)$