Why can I write $\frac{d}{dt}=\frac{d}{dt'}\frac{dt'}{dt}+\frac{d}{dx'}\frac{dx'}{dt}$? I’m dealing with a Lorentz invariance problem, and in one of the solutions I’ve seen to prove the wave equation the term above was used. However I don’t really understand why it can be written that way. Could someone provide an explanation?
 A: It's the chain rule for partial derivatives under the change of variables
$$
x= x(x',t')\\  
t= t'
$$ You need to be careful to specify what is being fixed in each derivative though, so it should be
$$
\left(\frac{\partial}{\partial t}\right)_x = \left(\frac {\partial}{ \partial t'}\right)_{x'}\left(\frac{\partial t'}{\partial t}\right)_x+ \left(\frac{\partial}{\partial x'}\right)_{t'} \left(\frac {\partial x'}{\partial t}\right)_{x},
$$
where
$$ 
\left(\frac{\partial t'}{\partial t}\right)_x=1
$$
A: For a function of two variables $\displaystyle f=f( x,y)$ the total differential is given by
\begin{equation*}
df=\frac{\partial f}{\partial x} dx+\frac{\partial f}{\partial y} dy.
\end{equation*}
The function $\displaystyle f$ depends explicitly only on $\displaystyle x$ and $\displaystyle y$. It can depend on more variables as well, but that dependence has to go through $\displaystyle x$ and $\displaystyle y$. If $\displaystyle x$ and $\displaystyle y$ depend on a variable $\displaystyle t$, we can find the derivative of $\displaystyle f$ with respect to $\displaystyle t$ by dividing the total differential by $\displaystyle dt$:
\begin{equation*}
\frac{df}{dt} =\frac{\partial f}{\partial x}\frac{dx}{dt} +\frac{\partial f}{\partial y}\frac{dy}{dt}
\end{equation*}
We can use this to define a differential operator that works on an arbitrary function of $\displaystyle x$ and $\displaystyle y$:
\begin{equation*}
\frac{d}{dt} =\frac{dx}{dt}\frac{\partial }{\partial x} +\frac{dy}{dt}\frac{\partial }{\partial y}
\end{equation*}
Let me know if there is something I can clear up.
