Q:

Second derivative symbol $\frac{d^2y}{dx^2}$!

I'm currently studying taking the second derivative of equations and I have been told the symbol used to represent second derivative is 

$$\frac{d^2y}{dx^2}$$

I was just wondering, why this symbol is chosen? Why is it not $$\frac{dy^2}{dx^2}$$ or $$\frac{d^2y}{d^2x}$$

A:

The first derivation is
$$\dfrac{dy}{dx}$$
taking other derivation respect to $x$ has the representation $\dfrac{d}{dx}$, then for second derivative we write
$$\dfrac{d}{dx}\left(\dfrac{dy}{dx}\right)=\dfrac{d^2y}{dx^2}$$
means in non-standard symbol 
$$\color{blue}{\dfrac{d\times d~y}{(dx)^2}}$$

A:

Differentiation by $x$ takes $y$ to
$$\frac{dy}{dx}$$
or if you like, to
$$\frac{d}{dx}y.$$
One would write, say
$$\frac{d}{dx}\cos x=-\sin x$$
etc.
So the derivative of $dy/dx$ is then
$$\frac{d}{dx}\frac{dy}{dx}$$
which one naturally abbreviates as
$$\frac{d^2y}{dx^2}.$$

