Finding inverse of a signal using analog circuits
If your variable is constrained to be > 0 you can use a simple log-antilog circuit which would likely cost a lot less than a purpose-built multiplier chip. For example, this one in an older TI app note. 



For the transistors you can use matched duals which cost less than 20 cents each, and the amplifier could be a quad op-amp. 

In this case E2 would be your input voltage. and E1 and E3 can be connected to a reference voltage.