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f ′ ( x ) = ( x + x + ⋯ + x ⏞ x t i m e s ) ′ = lim h → 0 { ( x + h ) + ( x + h ) + ⋯ + ( x + h ) ⏞ ( x + h ) t i m e s } − ( x + x + ⋯ + x ⏞ x t i m e s ) h = lim h → 0 ( h + h + ⋯ + h ) ⏞ x t i m e s + { ( x + h ) + ( x + h ) + ⋯ + ( x + h ) ⏞ h t i m e s } h = x + lim h → 0 ( x h + h 2 ) h = x + x = 2 x \begin{aligned}f'(x)&=(\overbrace{x+x+ \cdots +x}^{x\;\rm{times}})'\\&= \displaystyle\lim_{h \to \ 0} \frac {\{\overbrace{(x+h)+(x+h)+ \cdots +(x+h)}^{(x+h)\;\rm{times}}\} - (\overbrace{x+x+ \cdots +x}^{x\;\rm{times}})}{h}\\&= \displaystyle\lim_{h \to \ 0} \frac {(\overbrace{h+h+ \cdots +h)}^{x\;\rm{times}} + \{\overbrace{(x+h)+(x+h)+ \cdots +(x+h)}^{h\;\rm{times}}\}}{h}\\&=x +\displaystyle\lim_{h \to \ 0} \frac{(xh+h^2)}{h}\\&=x+x=2x\end{aligned} f ′ ( x ) = ( x + x + ⋯ + x x times ) ′ = h → 0 lim h { ( x + h ) + ( x + h ) + ⋯ + ( x + h ) ( x + h ) times } − ( x + x + ⋯ + x x times ) = h → 0 lim h ( h + h + ⋯ + h ) x times + { ( x + h ) + ( x + h ) + ⋯ + ( x + h ) h times } = x + h → 0 lim h ( x h + h 2 ) = x + x = 2 x lim x → a f ( x ) = L \displaystyle \lim_{ x \rightarrow a }{ f ( x ) } = L x → a lim f ( x ) = L