Question

What can be said about differentiation of a constant with respect to a constant & differentiation of a variable with respect to a constant?

Answer 1

Please see below.

Explanation:

One of the definition of the derivative of function `f` with respect to function `g` is

`(df)/(dg) = lim_(hrarr0) (f(x+h)-f(x))/(g(x+h)-g(x))`

If `g(x)` is a constant function, then `g(x+h) = g(x)` for all `x` and all `h`, so the denominator is always `0` and no limit exists.

This is true regardless of whether `f(x)` is constant or non-constant.

Another definition

`(df)/(dg) = (df)/dx * 1/((dg)/dx)`

But if `g` is constant then `(dg)/dx = 0`, so the second factor is not defined.
Again whether `f` is constant or variable, no derivative exists.