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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′=lim_(h→0) ((f(x+h)−f(x))/h)=lim_(h→0) (((x+h)^n −x^n )/h)       = lim_(h→0) ((Σ_(k=0) ^n C(n,k)x^(n−k)  h^k −x^n )/h)       = lim_(h→0) (( ((n),(1) ) x^(n−1) h+ ((n),(2) ) x^(n−2) h^2 +...+h^n )/h)       = lim_(h→0)  ((n),(1) ) x^(n−1) + ((n),(2) ) x^(n−2)  h +...+h^(n−1)         =  ((n),(1) ) x^(n−1)  = nx^(n−1)  .

$$\:{y}'=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{{n}} −{x}^{{n}} }{{h}} \\ $$$$\:\:\:\:\:=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}\left({n},{k}\right){x}^{{n}−{k}} \:{h}^{{k}} −{x}^{{n}} }{{h}} \\ $$$$\:\:\:\:\:=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:{x}^{{n}−\mathrm{1}} {h}+\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\:{x}^{{n}−\mathrm{2}} {h}^{\mathrm{2}} +...+{h}^{{n}} }{{h}} \\ $$$$\:\:\:\:\:=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:{x}^{{n}−\mathrm{1}} +\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\:{x}^{{n}−\mathrm{2}} \:{h}\:+...+{h}^{{n}−\mathrm{1}} \\ $$$$\:\:\:\:\:\:=\:\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:{x}^{{n}−\mathrm{1}} \:=\:{nx}^{{n}−\mathrm{1}} \:. \\ $$$$ \\ $$

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