Technology Musings

January 13, 2016

General / Ratios of Infinity

JB

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.

January 01, 2016

General / A Meaning for d^2y/d^2x?

JB

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

Or:

(d^2y/dx^2) / (d^2x/dy^2) * (dx/dy)^2

I will post an example later when I have a good one.