Question: The voltage generated in the transformer must be due to the change in the flux in the coils, but if the voltage is DC, why is no voltage induced?

Of course, we know that voltage is created by the derivative of flux, but what does it mean?

Answer: 
The reason we cannot use a transformer at DC is that a DC voltage would cause a linearly increasing flux in the core: \$\Phi\sim\frac{\mathrm d V}{\mathrm d t}\$. In reality, the core will [saturate](https://en.wikipedia.org/wiki/Saturation_(magnetic)) quickly, and the flux stops increasing. Then, also the current in the primary coil reaches a limit by the DC resistance of the winding.
Mathematically speaking, "derivative" means "rate of change". In *steady state*, DC does not change, the magnitude of the flux does not change, and the "derivative" = 0, so induced voltage is zero, and the "transformer" is reduced to the role of an electromagnet.