The cost of producing x ounces of gold from a new gold mine is C=f(x) dollars.

Do you think the values of f'(x) will increase or decrease in the short term? What about the long term? Explain.

decrease at start due to startup costs and paying off initial investment in staff and equipment

increase over long term because we have to go deeper and deeper into the vein as we mine out the easy stuff near ground level.

To determine whether the values of f'(x), the derivative of the cost function, will increase or decrease in the short term and long term, we need more information about the specific function f(x).

However, based on the given context, we can make some general observations.

In the short term:
- If the new gold mine is still in the early stages of production, it is likely that the cost of producing additional ounces of gold will initially decrease as the mine becomes more efficient in extracting gold.
- As the mine discovers and taps into richer gold deposits and gains more experience, they may be able to produce more gold per unit of cost, thus causing the derivative f'(x) to decrease or possibly even reach a minimum.
- However, in the later stages of production, once the easily accessible gold deposits are exhausted, it is possible that the cost of extracting additional ounces of gold may start to increase. This would imply that the derivative f'(x) would increase.

In the long term:
- It is expected that the cost of extracting gold from the mine will continue to increase as the mine explores deeper into the ground or adopts more costly extraction methods to reach gold reserves.
- As the mine progresses, it is likely that the derivative f'(x) will steadily increase over time, indicating that the cost for producing additional ounces of gold will rise.

To provide a more accurate analysis of whether f'(x) will increase or decrease in the short and long term, it would be necessary to have more specific information about the function f(x) and the characteristics of the gold mine in question.