That doesn't look right. I prefer to differentiate a trial function. Imagine we have a function ##f(t', x')##, where ##t' = \gamma(t - vx/c^2), \ x' = \gamma(x - vt)##.
Now, we differentiate ##f## with respect to ##x## using the chain rule:
##\frac{\partial f}{\partial x} = \frac{\partial f}{\partial t'}\frac{\partial t'}{\partial x} + \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial x} = \frac{\partial f}{\partial t'}(-\gamma v/c^2) + \frac{\partial f}{\partial x'}(\gamma)##
Now you need to take the second derivative of ##f## by repeating this process - and remember that you have to differentiate both terms using the chain rule, so you will get cross terms in ##\frac{\partial^2 f}{\partial t' \partial x'}##.
At the end, you can remove the trial function to leave, for example:
##\frac{\partial}{\partial x} = (-\gamma v/c^2)\frac{\partial }{\partial t'} + (\gamma)\frac{\partial}{\partial x'}##
That's what you get if you only wanted the first derivative.
Note that you can do the chain rule on differentials, but I prefer to have a function to differentiate, and then take the function away when I'm finished.
