Mixed partial derivatives are read from right to left by convention.
Partial derivatives can be taken using the normal procedures for single-variable
calculus if you treat the constant variables as such
Example: Taking partial derivatives
For the function \(f(x,y) = x^2 \cos y\), evaluate \((\partial f/\partial x)_y\),
\((\partial^2 f/\partial x^2)_y\), and \(\partial^2 f/\partial y \partial x\).
The first partial derivative with respect to x (treating y as a constant) is
(2.4)#\[\begin{equation}
\td{}{f}{x}{y} = 2 x \cos y
\end{equation}\]
The second (repeated) partial derivative with respect to x (again, treating
y as a constant) is
(2.5)#\[\begin{equation}
\td{2}{f}{x}{y} = \pp{}{}{x}\left[ \td{}{f}{x}{y} \right]_y
= \pp{}{}{x}\left( 2 x \cos y \right)_y = 2 \cos y
\end{equation}\]
The second (mixed) partial derivative with respect to x then y is
(2.6)#\[\begin{equation}
\frac{\partial^2 f}{\partial y \partial x} = \pp{}{}{y}\left[ \td{}{f}{x}{y} \right]_x
= \pp{}{}{y} \left( 2 x \cos y \right)_x = -2x \sin y
\end{equation}\]
Example: Order of mixed second partial derivatives
Show the order of the mixed derivatives does not matter for the example function
given above.
The first partial derivative with respect to \(y\) is
(2.8)#\[\begin{equation}
\td{}{f}{x}{y} = -x^2 \sin y
\end{equation}\]
So, the second (mixed) partial derivative with respect to y then x is
(2.9)#\[\begin{equation}
\frac{\partial^2 f}{\partial x \partial y} = \pp{}{}{x}\left[ \td{}{f}{y}{x} \right]_y
= \pp{}{}{x} \left( -x^2 \sin y \right)_y = -2x \sin y
\end{equation}\]