Question-166649

Question Number 166649 by kurt last updated on 24/Feb/22

Commented by greogoury55 last updated on 27/Feb/22

y^{'} = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h} = \lim_{h \to 0} \frac{{(x + h)}^{n} - x^{n}}{h}

= \lim_{h \to 0} \frac{\underset{k = 0}{\sum^{n}} C (n, k) x^{n - k} h^{k} - x^{n}}{h}

= \lim_{h \to 0} \frac{(\begin{matrix} n \\ 1 \end{matrix}) x^{n - 1} h + (\begin{matrix} n \\ 2 \end{matrix}) x^{n - 2} h^{2} + \dots + h^{n}}{h}

= \lim_{h \to 0} (\begin{matrix} n \\ 1 \end{matrix}) x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) x^{n - 2} h + \dots + h^{n - 1}

= (\begin{matrix} n \\ 1 \end{matrix}) x^{n - 1} = {n x}^{n - 1} .