Ideas like differentiation could help show how electricity and magnetism are related. In fact, physicists actually look at them as a single force called electromagnetism. But, it is important to note that a magnet not moving or changing relative to an electron will not put any force on the electron. Said another way: an unchanging magnetic field will not create or change an electric field. The reverse is also true: an unchanging electric field will not create or change a magnetic field. This change is required both ways at the same time.
Your point is a critical idea in physics, a self sustaining electromagnetic field due to constant change. The idea is explained in Maxwell's equations. The idea is used as the explanation of light, also known as electromagnetic radiation. The idea is also used in the explanation of the earths magnetic field. And in the explanation of the newly developing model of a Polariton. And most likely in many more areas of science.
If you want to know more about electromagnetism, and you already know how differentiation and force fields work, I would recommend starting with Maxwell's equations, then move into understanding Polarization.
It appears people may be upset about the lack or presence of causality in my answer (a comment after a down vote would be nice)... I would like to address this point, and the comment provided, but it has no simple answer.
If I use a magnet in a generator to make electricity the cause is clear. But, then I can use that same motor to do the opposite, generating motion (a changing magnetic field) from electricity. Clearly one is not the cause of the other.
A case for causality could certainly be made for earth's magnetic field and core current... but which one actually did come first the chicken or the egg. Not sure geologists have enough evidence to prove a changing magnetic or changing electric field came first, just knowledge that our magnetic poles swap.
From the case of earth it is clear the question is not easily answerable. In quantum physics the electric field of an electron must have an angular momentum to give it the magnetic field measured in experiments. However, it does not actually spin.
The electric and magnetic fields themselves do not experience the concept we know as causality, only the transfer of momentum using the electromagnetic force is a result of causality.