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 $$\frac{d^2y}{d^2x}$$
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 $$\color{blue}{\dfrac{d\times d~y}{(dx)^2}}$$
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 $$\frac{d^2y}{dx^2}.$$
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 $\frac{d^2f}{(dx)^2}$.
$d^2y$ 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.
$$ y'=\frac {d}{dx}(y)=\frac {dy}{dx} $$
$$ y''=\frac {d}{dx}(\frac{d}{dx}y)=\frac {d^2y}{dx^2} $$ $$ y'''=\frac {d}{dx}(\frac {d^2}{dx^2}(y))=\frac {d^3y}{dx^3}$$
