Step 1: Define the function. Let $f(x) = x^2$. Step 2: Write down the formula for the derivative using the first principle. $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ Step 3: Find $f(x+h)$. Substitute $(x+h)$ into the function $f(x)$: $$ f(x+h) = (x+h)^2 $$ Expand $(x+h)^2$: $$ (x+h)^2 = x^2 + 2xh + h^2 $$ Step 4: Substitute $f(x+h)$ and $f(x)$ into the first principle formula. $$ f'(x) = \lim_{h \to 0} \frac{(x^2 + 2xh + h^2) - x^2}{h} $$ Step 5: Simplify the numerator. $$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$ Step 6: Factor out $h$ from the numerator. $$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$ Step 7: Cancel out $h$ (since $h \neq 0$ as $h \to 0$). $$ f'(x) = \lim_{h \to 0} (2x + h) $$ Step 8: Evaluate the limit by substituting $h=0$. $$ f'(x) = 2x + 0 $$ $$ f'(x) = 2x $$ The derivative of $x^2$ using the first principle is: $$ \boxed{2x} $$