Depends on your point of view ...
I would say the op-amp is a bog standard inverting amp.
To calculate the gain you need to know the input impedance (R1 in parallel with the mic impedance, all in series with the impedance presented by C1), if this is glossed over then you could say that the sum of R2 & R3 set the gain (but strictly speaking you need that input impedance to find the ratio).
That cap, C1 is the differentiator, if there really is one, in combination with its series resistor, the impedance of the mic.
R4 & R5 set the op amp's bias, the point it sits at when there is no other input, in my mind a threshold is only used to determine a switching point so I wouldn't agree with this being called that.
C2 is an AC decoupling cap but cannot be considered without R6 & R7 (cropped off your diagram), it stops the DC bias of one circuit stage upsetting the next stage (in this case a 555 monostab trigger input). 1/2VCC on one side, 2/5VCC on the other.
You could also argue that C2 and its series resistance is a differentiator.
The thing with these "differentiators" is their intended purpose.
In my mind the caps are just for DC blocking as Kinjal said. If the mic gave a step-change in DC output then a differentiator would 'do its thing' but all we want here is to pass mid-to-high audio frequencies such that the amp can, given a loud enough input, trigger the monostab.
All that matters here is volume. To prove this the input stage could be redesigned to use DC coupling and thus eliminate C1, or C1 could be swapped for 100uF, if a differentiator is reqired that would stop the circuit working (which it won't !).