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

Commented by greogoury55 last updated on 27/Feb/22

$$y^{\prime }=\underset{h\to 0}{\lim }\frac{f(x+h)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{{(x+h)}^n-x^n}{h}$$$$=\underset{h\to 0}{\lim }\frac{\underset{k=0}{\sum ^n}C(n,k)x^{n-k}{h}^k-x^n}{h}$$$$=\underset{h\to 0}{\lim }\frac{\left(\begin{array}{c}n\\ 1\end{array}\right)x^{n-1}h+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}{h}^2+...+h^n}{h}$$$$=\underset{h\to 0}{\lim }\left(\begin{array}{c}n\\ 1\end{array}\right)x^{n-1}+\left(\begin{array}{c}n\\ 2\end{array}\right)x^{n-2}h+...+h^{n-1}$$$$=\left(\begin{array}{c}n\\ 1\end{array}\right)x^{n-1}={nx}^{n-1}.$$$$$$

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