Why dx squared (dx)² = 0？

Let’s take the derivative of

• 𝑓(𝑥) = 𝑥²

The definition of derivative:

• $\frac{df}{dx}=\frac{\text{amount of change in output}}{\text{amount of change in input}}$
• $\frac{df}{dx}={\lim }_{dx\to 0}\frac{f(x+dx)-f(x)}{dx}$
• $\frac{df}{dx}={\lim }_{dx\to 0}\frac{(x+dx{)}^2-x^2}{dx}$
• $\frac{df}{dx}={\lim }_{dx\to 0}\frac{x^2+2xdx+d{x}^2-x^2}{dx}$
• $\frac{df}{dx}={\lim }_{dx\to 0}\frac{2xdx+d{x}^2}{dx}$
• $\frac{df}{dx}={\lim }_{dx\to 0}2x+dx$
• $\frac{df}{dx}=2x$