Calc2.60

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f(x)=x^{n}\,

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)^{n}-x^{n}}{\Delta x}}

=\lim _{{\Delta x\to 0}}{\frac  {x^{n}+nx^{{n-1}}\Delta x+{\frac  {n(n-1)}{2!}}x^{{n-2}}\left(\Delta x\right)^{2}+...+\left(\Delta x\right)^{n}-x^{n}}{\Delta x}}

=\lim _{{\Delta x\to 0}}\left[nx^{{n-1}}+{\frac  {n(n-1)}{2!}}x^{{n-2}}\left(\Delta x\right)+...+\left(\Delta x\right)^{{n-1}}\right]

=nx^{{n-1}}\,