Calc2.54

From Exampleproblems

$f(x)=x\,$

$f'(x)=\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}=\lim_{\Delta x \to 0}\frac{x+\Delta x-x}{\Delta x}=\lim_{\Delta x \to 0}\frac{\Delta x}{\Delta x}=\lim_{\Delta x \to 0}1=1$

Since the limit exists, we have shown that the derivative of $x\,$ is 1.