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Marginal cost = derivative of (Total cost/Quantity)

Where Total cost = fixed cost + variable cost

Marginal cost = derivative (Variable cost/Quantity) (by definition, fixed costs do not vary with quantity produced)

Average cost = Total cost/Quantity

The rate of change of average cost is equivalent to its derivative.

Thus,

AC' = derivative(Total cost/Quantity) => derivative (Variable cost/Quantity) = MC.

So, when MC is increasing, AC' is increasing. That is, when marginal cost increases, the rate of change of average cost must increase, so average cost is always increasing when marginal cost is increasing.

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