Partial derivatives and Galilean transforms

Question

I have trouble understanding the following:

\begin{equation} \frac{\partial}{\partial x'} = \frac {\partial x}{\partial x'} \frac{\partial}{\partial x} + \frac{\partial t}{\partial x'} \frac{\partial}{\partial t} \\ \frac{\partial}{\partial t'} = \frac {\partial x}{\partial t'} \frac{\partial}{\partial x} + \frac{\partial t}{\partial t'} \frac{\partial}{\partial t} \end{equation}

given that $x = x' -vt$ and $t=t'$. I have a mental block in how the expression above are derived. Can someone give a dumb down explanation?

Answer 1

This is the chain rule for derivatives.

Answer 2

Just apply the differential operators to an arbitrary function of $(t,x)$ computed at $t(t',x')$ and $x(t',x')$; then use chain differentiation:

$$ \frac{\partial}{\partial t'}\,f(t(t',x'),x(t',x'))=\frac{\partial f}{\partial t}(t(t',x'),x(t',x'))\,\frac{\partial t}{\partial t'}(t',x')+\frac{\partial f}{\partial x}(t(t',x'),x(t',x'))\,\frac{\partial x}{\partial t'}(t',x') $$

and similarly for $x'$. Since this is true of all functions $f$, the relations you wrote hold.