Derivation of Normal Distribution

The derivation can be found here. There is so much to learn from this proof. You can try to read it to see if you understand it. If you encounter difficulty understanding the first part of the proof, you can refer to the explanations below.

Concepts:
integration as sum of area of rectangles
probability as area under the curve
describing one thing from different perspectives
polar coordinates
polar integration
solving differential equation by separating variables
differential equation true for any $x$ and $y \Rightarrow$ equals a constant
large errors less likely than small errors $\Rightarrow C$ must be negative
characteristic of a probability distribution: total area under the curve must be $1$
symmetry: properties of even and odd functions
rewriting the product of two integrals as a double integral
evaluating a double integral using polar coordinates
improper integral
integration by parts
transformation of graphs

Explanations:

The probability of the dart falling in the vertical strip from $x$ to $x+\Delta x$ is $p(x)\Delta x$. We can think of probability as the area under the curve. $p(x)\Delta x$ is the area of the rectangle. Similarly, the probability of the dart falling in the horizontal strip from $y$ to $y+\Delta y$ is $p(y)\Delta y$.

Since we assumed that errors in perpendicular directions are independent, the probability of the dart falling in the shaded region can be given by $\large p(x)\Delta x \cdot p(y) \Delta y$. Now, any region $r$ units from the origin with area $\Delta x \cdot \Delta y$ has the same probability. [Finding probability from two perspectives] We thus have $\large p(x)\Delta x \cdot p(y) \Delta y=g(r)\Delta x \Delta y$, which means $g(r)=p(x)p(y)$.

[Sidenote: Using two methods to express the same thing can yield interesting results.
$\int \sin 2x \cos x dx=\int 2\sin x \cos^2 x dx=-2\int u^2 du=-\frac{2}{3}\cos^3 x+C_1$
$\int \sin 2x \cos x dx=\frac{1}{2}\int(\sin 3x+\sin x)dx=-\frac{1}{6}\cos 3x-\frac{1}{2}\cos x+C_2$
$\frac{-2}{3}\cos^3 x=-\frac{1}{6}\cos 3x-\frac{1}{2}\cos x+K$
Put $x=0$, we have $K=0$.
Therefore, we reach the identity $\cos 3x=4\cos^3 x-3\cos x$ from integration!]

The remainder of the proof is well-explained in the link.

Related:
Same derivation with some details
A similar derivation
Another similar derivation
Relation to central limit theorem

More to explore:
Derivation of Gaussian Distribution from Binomial
Deriving the normal density as a solution of a differential equation
Other advanced derivations
Distributions Derived from the Normal Distribution