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Question Number 22105 by j.masanja06@gmail.com last updated on 11/Oct/17
use the first principle to find  value of  f(x)=(x)^(1/3)
$${use}\:{the}\:{first}\:{principle}\:{to}\:{find} \\ $$$${value}\:{of} \\ $$$${f}\left({x}\right)=\left({x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$
Answered by $@ty@m last updated on 11/Oct/17
(dy/dx)=lim_(δx→0) ((f(x+δx)−f(x))/(δx))  =lim_(x+δx→x) (((x+δx)^(1/3) −x^(1/3) )/((x+δx)−x))  =(1/3)x^((1/3)−1) , using formula lim_(x→a) ((x^n −a^n )/(x−a))=na^(n−1)   =(1/(3x^(2/3) ))Ans.
$$\frac{{dy}}{{dx}}=\underset{\delta{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+\delta{x}\right)−{f}\left({x}\right)}{\delta{x}} \\ $$$$=\underset{{x}+\delta{x}\rightarrow{x}} {\mathrm{lim}}\frac{\left({x}+\delta{x}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} −{x}^{\frac{\mathrm{1}}{\mathrm{3}}} }{\left({x}+\delta{x}\right)−{x}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}{x}^{\frac{\mathrm{1}}{\mathrm{3}}−\mathrm{1}} ,\:{using}\:{formula}\:\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{n}} −{a}^{{n}} }{{x}−{a}}={na}^{{n}−\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}{x}^{\frac{\mathrm{2}}{\mathrm{3}}} }{Ans}. \\ $$

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