How is the second derivitive derived? As everyone knows that the derivitive of a function is notated as $\frac{dy}{dx}$

The question is:
How is the second derivitive $\left(\frac{d^2y}{dx^2}\right)$ notation derived?
This is just notation. Let $y=f(x)$. Then $\frac d{dx}(y)=f'(x)$.  Now
$$f''(x)=\frac d{dx}\left(\frac d{dx}(y)\right)=\left(\frac d{dx}\right)^2(y)=\frac {d^2}{dx^2}(y).$$
Edit: I misread the OP question and thought that practical examples were asked for.

The second derivative arises from considering the first derivative as an independent function, and is obtained by derivation.
Notation (like the other answerer mentioned), is mainly of two types:
Leibniz notation or:
$ {d\over dx} (f(x))$ is the first order derivative
${d\over dx} ({d\over dx}(f(x)) = {d^2\over dx}f(x)$

Lagrange Notation instead uses "prime" marks:
Derivative of $f(x) = f'(x)$, derivative of $f'(x) = f''(x)$ and so on.

Practical example:
$$
f(x) = x^3
$$
$$
f'(x) = {d\over dx}f(x) = 3x^2
$$
$$
f''(x) = {d\over dx}f'(x) = 6x
$$
A pratical example is however the change in kinetic energy of a moving body.