What do the derivative or the integral of amplitude of a seismogram mean?
This is a very simple answer, and it depends on the what the seismometer is measuring. I'm assuming you mean the time derviative/intdegral  $\frac\partial{\partial{}t}$ or  $\int{dt}$

Most seismometers measure displacement over time or velocity over time (series). So taking the time derivative will give you the spectral velocity or the spectral acceleration respectively when you take their derivatives.

When you take the integral of displacement, you calculate the total area displaced over time, and when you take the integral of velocity, you calculated the displacement. 

It's all physics 101 really!