Find the derivation of the following with respect to x using first principle and find the value of the following limits.

The derivative $F'(x)$ is defined as:

$$F'(x) = \lim_{h \to 0} \frac{F(x+h) - F(x)}{h}$$

Given $F(x) = x^2 - x$, we find $F(x+h) = (x+h)^2 - (x+h)$.

$$F'(x) = \lim_{h \to 0} \frac{((x+h)^2 - (x+h)) - (x^2 - x)}{h}$$

Expand $(x+h)^2$:

$$F'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2 - x - h) - (x^2 - x)}{h}$$

Distribute the negative sign and combine like terms in the numerator:

$$F'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x - h - x^2 + x}{h}$$ $$F'(x) = \lim_{h \to 0} \frac{2xh + h^2 - h}{h}$$

$$F'(x) = \lim_{h \to 0} \frac{h(2x + h - 1)}{h}$$

Since $h \to 0$ but $h \neq 0$, we can cancel $h$:

$$F'(x) = \lim_{h \to 0} (2x + h - 1)$$

$$F'(x) = 2x + 0 - 1$$ $$F'(x) = \boxed{2x - 1}$$