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De afgeleide van xⁿ

\[\begin{array}{l} f(x) = {x^n}\\ f'(x) = \\ \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^n} - {x^n}}}{h} = \\ \mathop {\lim }\limits_{h \to 0} \frac{{\left( {\begin{array}{*{20}{c}} n\\ 0 \end{array}} \right){x^n} + \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right){x^{n - 1}}h + \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^{n - 2}}{h^2} + ...\left( {\begin{array}{*{20}{c}} n\\ n \end{array}} \right){h^n} - {x^n}}}{h} = \\ \mathop {\lim }\limits_{h \to 0} \frac{{\left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right){x^{n - 1}}h + \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^{n - 2}}{h^2} + ...\left( {\begin{array}{*{20}{c}} n\\ n \end{array}} \right){h^n}}}{h} = \\ \mathop {\lim }\limits_{h \to 0} \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right){x^{n - 1}} + \left( {\begin{array}{*{20}{c}} n\\ 2 \end{array}} \right){x^{n - 2}}{h^1} + ...\left( {\begin{array}{*{20}{c}} n\\ n \end{array}} \right){h^{n - 1}} = \\ \left( {\begin{array}{*{20}{c}} n\\ 1 \end{array}} \right){x^{n - 1}} = \\ n \cdot {x^{n - 1}} \end{array}\]

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