# Confusion in the difference between DC current gain and AC current gain in common emitter amplifier

Here is a definition:

It is easy to understand what DC current gain means here. The base current is constant, the collector current is also constant. An their ratio βdc = Ic/Ib. This is the DC current gain.

But I don't get the meaning of the βac AC current gain formula here. Is that the derivative of Ic wrt Ib? I came across the following plot but still didn't get how to calculate this gain.

How would one practically calculate this if one knows the input and output currents? Imagine the base current Ib is a sinusoidal current signal with 10uA peak to peak; and Ic is 1mA peak to peak. If it was DC current we would say the beta is 100. But here since we are dealing with AC, we need to find the AC gain by looking at the Ic Vce characteristics. How can we progress and make a logical derivation of AC current gain from this point?