A lot of people do a lot of thinking about whether or not infinities exist, and wonder how real numbers can add up to infinity.
Sometimes I wonder if that sort of thought might be the inverse of reality - perhaps the infinities are the foundational realities, and all finite numbers are merely infinities in ratio.
I have recently been randomly curious about random calculus-y things. Anyway, the notion for the second derivative of a function is d^2y/dx^2. This is the second differential of y divided by the first differential of x squared.
However, there is, technically, also a d^2x, though it doesn't get much attention. And, d^2y and d^2x can be put into ratio with each other, but I don't really know what it means. But it is an interesting operation nonetheless.
So, the derivative of an equation is dy/dx and the derivative of its inverse is dx/dy.
The second derivative of an equation is d^2y/dx^2 and the second derivative of its inverse is d^2x/dy^2.
Therefore, to get the d^2y/d^2x you just do:
(2nd Derivative of y wrt x / 2nd derivative of x wrt y) * (First derivative of the inverse)^2
(d^2y/dx^2) / (d^2x/dy^2) * (dx/dy)^2
I will post an example later when I have a good one.