Where do I go wrong with the product rule with three variables?

Question

I am not sure how to use the product rule for differentiation, when I have three (or more) variables. For example, how would I solve this?

$$(y \frac{d}{dz} - z \frac{d}{dy}) (z \frac{d}{dx} - x \frac{d}{dz})$$

Because I thought it goes like this (For example, just for the $y \frac {d}{dz} * z \frac{d}{dx}$ part):

You first multiply everything, so you get:

$$ yz \frac{d}{dz} \frac{d}{dx}$$ and then get the sum of the individual derivatives, like: $$yz \frac{d}{dz} + yz \frac{d}{dx}$$

But it's wrong and I don't know why, where did I go wrong?

Edit: I didn't realise you need a function, but I am not sure how to answer, so here is the whole example (The title says: Calculate the commutator): The whole problem

Answer 1

The differential operator $\dfrac {\partial}{\partial x}$ acts on functions via differentiation, on other differential operators via composition, and obeys the product rule. For instance, $$\frac{\partial}{\partial z} \left( z \frac{\partial}{\partial x} \right) = \frac{\partial z }{\partial z} \frac{\partial}{\partial x} + z\frac{\partial}{\partial z}\frac{\partial}{\partial x} = \frac{\partial}{\partial x} + z \frac{\partial^2}{\partial z \partial x}$$ so that $$\left( y \frac{\partial}{\partial z} \right)\left( z \frac{\partial}{\partial x} \right) = y \frac{\partial}{\partial x} + yz \frac{\partial^2}{\partial z \partial x}$$

Answer 2

You are making three mistakes.

First, you assume that $yz\frac{\partial}{\partial z}\frac{\partial}{\partial x}$ is the same as $y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}$. Let's put a simple function $w(x,y,z)=x$ into the mix to see why this is wrong:

$yz\frac{\partial}{\partial z}(\frac{\partial w}{\partial x}) = yz\frac{\partial}{\partial z}(1)=0$

$y\frac{\partial}{\partial z}(z\frac{\partial w}{\partial x})=y\frac{\partial}{\partial z}(z)=y$

In fact, by the product rule, $y\frac{\partial}{\partial z}z\frac{\partial}{\partial x}$ is equal to $y(\frac{\partial z}{\partial z}\frac{\partial}{\partial x}+yz\frac{\partial}{\partial z}\frac{\partial}{\partial x})=y(\frac{\partial}{\partial x}+z\frac{\partial^2}{\partial z\partial x})$

Second, $\frac{\partial}{\partial z}\frac{\partial}{\partial x}$ is not equal to $\frac{\partial}{\partial z}+\frac{\partial}{\partial x}$. Why would it be?

Third: $\frac{d}{dz}$ is not the same as $\frac{\partial}{\partial z}$!