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