Understanding the Impact of Difficulty on Mining Efficiency

I've been saying that if Difficulty rises by 10%, miner productivity drops by about 10%. Seems like a decent rule of thumb, but it doesn't hold up for bigger changes. For instance, if Difficulty jumps by 100%, miner productivity won't just drop to zero; it actually decreases by 50% instead. So how do we accurately describe the percentage decrease in productivity when Difficulty increases by a certain percentage? I’m not a math whiz or anything, but I can grasp a formula if it’s laid out clearly.

Lets say you earn $100 a day and it costs you $50 a day to earn that. If your earning drops 10% your profit drops 20% ... i.e. the cost of earning makes a big difference