I'm trying to learn to derive the pdf of normal distribution, that is
$$f(x; \mu,\sigma^2)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\left[−\dfrac{1}{2}\left(\dfrac{x−\mu}\sigma\right)^2\right]$$
Firstly, there is a hypothesis to derive this pdf, which is the coordinate is farther away from the origin, the lower the value of $f(x)$.
After defining the area and transformation to polar form on the Cartesian plane, I get $f(x)=Ae^{\frac{Cx^2}2}$.
By the hypothesis, the C of $f(x)=Ae^{\frac{Cx^2}2}$ will be negative. So the question is what is the theorem that implies C is negative
That the pdf integrates to $1$ instead of $\infty$.