High School Calculus/The Derivative

The definition of a Derivative of a Function $$f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{f(x+{\Delta}x)-f(x)}{{\Delta}x}$$ Example $$f(x)=x^2$$Use the limit definition with the given function $$f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{(x+{\Delta}x)^2-x^2}{{\Delta}x}$$ $$f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{(x^2+2x{\Delta}x+{\Delta}x^2)-x^2}{{\Delta}x}$$ $$f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{2x{\Delta}x+{\Delta}x^2}{{\Delta}x}$$ $$f'(x)=\lim_{{\Delta}x\rightarrow 0} \frac{{\Delta}x(2x+{\Delta}x)}{{\Delta}x}$$ $$f'(x)=\lim_{{\Delta}x\rightarrow 0} (2x+{\Delta}x)$$ $$f'(x)=2x+0$$ $$f'(x)=2x$$