Partial Derivative.

Question

Can I write the right hand side of the following equation ?

$$\frac{\partial}{\partial x}[\frac{\partial f(y)}{\partial y}]=\frac{\partial^2 f(y)}{\partial y\partial x}.$$

Does this operator follow the property ?

Answer 1

If $f$ carries an open interval of $\mathbb{R}$ to $\mathbb{R}$, then it makes no sense to find a partial derivative of $f$; by definition a partial derivative of a map is the directional derivative of the map in the direction of a standard unit vector, provided that the map is a scalar field.

If $n \geq 2$, if $f$ carries some set $A$ open in $\mathbb{R}^{n}$ to $\mathbb{R}$, and if $f$ is twice differentiable on $A$, then we have $$ D_{i}D_{j}f = D_{j}D_{i}f $$ on $A$ for all $1 \leq i,j \leq n$; this is an elementary result in analysis. In particular, if $n := 2$, then $D_{1}D_{2}f = D_{2}D_{1}f$ on $A$; in the language of applied math, we may write this as $$ \frac{\partial^{2}f}{\partial x \partial y} = \frac{\partial^{2}f}{\partial y \partial x}, $$ which is of course valid on $A$ too.

Answer 2

If $f$ is a (differentiable) function of $y$ alone, then of course the partial derivative with respect to any other variable will be zero.