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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  {2(x+\Delta x)^{3}-2x^{3}}{\Delta x}}

=\lim _{{\Delta x\to 0}}{\frac  {2(x^{3}+3x^{2}\Delta x+3x\Delta x^{2}+\Delta x^{3})-2x^{3}}{\Delta x}}

=\lim _{{\Delta x\to 0}}{\frac  {6x^{2}\Delta x+6x\Delta x^{2}+2\Delta x^{3}}{\Delta x}}

=\lim _{{\Delta x\to 0}}6x^{2}+6x\Delta x+2\Delta x^{2}=6x^{2}

Since the limit exists, we have shown that the derivative of 2x^{3} is 6x^{2}.

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