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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.

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