Where do Maxwell's equations come from? I recently started learning the basic forms (integrals) of the Maxwell's equations, and everything that is related to electromagnetism seems to be derived from these fundamental equations. Now my question is, where did these equations come from and could you derive them (without using the strange inverted delta sign, yes, im not that far in mathematics yet) using basic calculus? Are these equations based on principles of conservation?
 A: The answer lies in another wiki article, 

In electromagnetism, one of the fundamental fields of physics, the introduction of Maxwell's equations (mainly in "A Dynamical Theory of the Electromagnetic Field") was one of the most important aggregations of empirical facts in the history of physics. It took place in the nineteenth century, starting from basic experimental observations, and leading to the formulations of numerous mathematical equations, notably by Charles-Augustin de Coulomb, Hans Christian Ørsted, Carl Friedrich Gauss, Jean-Baptiste Biot, Félix Savart, André-Marie Ampère, and Michael Faraday. The apparently disparate laws and phenomena of electricity and magnetism were integrated by James Clerk Maxwell, who published an early form of the equations, which modify Ampère's circuital law by introducing a displacement current term. He showed that these equations imply that light propagates as electromagnetic waves.

Go to the  link to find the original links  for each of these empirical laws.
Maxwell's tying up the various laws gathered from experimental observations and the verification of this most inclusive formulation with new observations and measurements established the theory of classical electrodynamics on par with classical mechanics.  thermodynamics  and statistical mechanics .  It was a beautiful set of theories .
Lord Kelvin was supposed to say 

There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

(The link has a caveat that no solid reference has been found though.)
It was a belief held before the advent of x-rays , the photoelectric effect and black body radiation discrepancies.
It does set the frame for the unexpected appearance of data that led to the quantum mechanical revolution.
A: Where do Maxwell's equations come from?
Oliver Heaviside. They aren't Maxwell's equations. What Anna referred to about Coulomb, Ørsted, Gauss, Biot, Savart, Ampère, and Faraday is all well and good, but see where it says this in the Wikipedia article:
"The powerful and most widely familiar form of Maxwell's equations, whose formulation is due to Oliver Heaviside, in the vector calculus formalism, is used throughout unless otherwise explicitly stated."
What's presented to you as Maxwell's equations are in some respects misleading. Again from the Wikipedia article: 
"The equations introduce the electric field E, a vector field, and the magnetic field B, a pseudovector field, where each generally have time-dependence".
However Maxwell unified electricity and magnetism to deliver the electromagnetic field, see this article for more. Which is why in Jackson's Classical Electrodynamics you can read "one should properly speak of the electromagnetic field Fμν rather than E or B separately". However this just doesn't feature in "Maxwell's equations". When you read the original material it's actually very different to what's generally taught, and I think some important meaning has been lost. The vector formalism describes what they do rather than what they are and why they do what they do. Which is why you will struggle to find a depiction of the electromagnetic field Fμν.   
