Why are the uncertainties so different? Here is my scenario: I am trying to calculate the uncertainty of the function $y=x^2$, that is, I want to find $\Delta y$, and I found that we can get a great difference in the $\Delta y$, depending on the method that is used.

The first approach is as following:

$$\frac{ \Delta y}{y}=2\times\frac{ \Delta x}{x} \implies \Delta y = 2\times \frac{ \Delta x}{x} \times y$$

The second method is as following:

$$\frac{ \Delta y}{\Delta x} =| \frac{dy}{dx} | \implies \Delta y= | \frac{dy}{dx} | \Delta x = 2| x | \Delta x$$

Mathematically, the difference I can see is that in the first example,
$$ \Delta y = 2\times \frac{ y}{x} \Delta x $$
we look at the average change in $y$ and change in $x$.

My question is, which of the following methods can be said to be more correct?
Second method is right because when you are doing |dy/dx| it is actually saying Δy/Δx limits to 0, which in fact will give more precise answer.
Second method is right because when you are doing |dy/dx| it is actually saying Δy/Δx limits to 0, which in fact will give more precise answer.