second order partial derivative notation What is the reason for a second-order derivative notation to be:  

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

and not:  

$\frac{d^2y} {d^2x}$ or $\frac{dy^2} {dx^2}$

Thanks in advance!
Suppose you want to take the derivative of 
$\newcommand{d}{\mathrm d}f(x) = \left(e^x + \frac1x\right).$
You can write

$$ \frac\d{\d x} \left(e^x + \frac1x\right). $$

If you want the second derivative of $f(x)$, that's the derivative of the derivative:

$$ \frac\d{\d x} \left( \frac\d{\d x} \left(e^x + \frac1x\right)\right). $$

Now it's a notational convention that you can write $\d^2$ to combine the two $\d$ symbols in the "numerator" and $\d x^2$ to combine the two $\d x$ symbols in the "denominator".
But it's just a convention.

It might be a little clearer if we wrote $(\d x)^2$ instead of $\d x^2,$
but (again) it's just a notational convention anyway;
we are not really multiplying $\d x$ by $\d x.$
The first derivation to any equation Let it be (y) is as follows:

d\dx (y)

When we take the second derivative of the function (y) , We must hit the first differential at :

(d\dx) * ( d\dx (y) )

Finally, the second derivative becomes the following :

( d^2\dx^2 (y) )

( d^2(Y)\dx^2 )

I wish God success .