How does inductor resist the increasing dc current?
You said it in your question: "an increase in DC current". Sometimes people get all bundled up in the term "AC". AC describes a special kind of changing current which is sinusoidal. We know pretty much everything there is to know about sinusoids, so describing the nature of AC is pretty easy.

An increase in DC current is still a changing current just like the current changes in an AC signal. The same magnetic effects apply (per your question) when there is any type of change in current. It just so happens that AC is a special type of changing current.

Having said that, describing the true conditions of the inductor and its related fields under arbitrarily changing currents requires more general use of electrodynamics laws.

If you want some more elaboration, please speak up. We'll talk about frequency and the Fourier representation of signals to go in a little deeper.