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
$$\frac{d^2y}{dx^2}.$$
A: If you think of the first derivative $g=\frac{df}{dx}$ as a function you would like to differentiate, the result should have the symbol $\frac{dg}{dx}=\frac{d\frac{df}{dx}}{dx}$. The latter is naively "computed" to be $\frac{d^2f}{(dx)^2}$.
A: $d^2y$ expresses that we have an (infinitesimal) second order difference, the difference of a difference, while $dx^2$ is indeed the square of an (infinitesimal) first order.
Might have been clearer to write $\dfrac{d_2y}{dx^2}$ instead.
A: $$ y'=\frac {d}{dx}(y)=\frac {dy}{dx} $$
$$ y''=\frac {d}{dx}(\frac{d}{dx}y)=\frac {d^2y}{dx^2} $$
$$ y'''=\frac {d}{dx}(\frac {d^2}{dx^2}(y))=\frac {d^3y}{dx^3}$$
