The derivative of 1 with respect to x is zero, since the derivative of any constant function is zero.

So you must be saying that the derivative of 1/f(x) with respect to x is zero for any f(x), where f is a differentiable function, with non-vanishing derivative near x (for it to be defined in the first place).

That doesn't make any sense.

Please don't respond to this. It's getting absurd.