Question Number 123147 by aurpeyz last updated on 23/Nov/20
$${from}\:{first}\:{principle}\:{obtain}\:{the}\: \\ $$$${diffrential}\:{coefficient}\:{of}\:{cos}\left({x}\right) \\ $$
Answered by TANMAY PANACEA last updated on 23/Nov/20
$$\frac{{dy}}{{dx}}=\:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{cos}\left({x}+{h}\right)−{cosx}}{{h}} \\ $$$$\frac{{d}\left({cosx}\right)}{{dx}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{sin}\left({x}+\frac{{h}}{\mathrm{2}}\right)}{\frac{{h}}{\mathrm{2}}×\mathrm{2}}×\left(−{sin}\frac{{h}}{\mathrm{2}}\right) \\ $$$$={sin}\left({x}+\frac{\mathrm{0}}{\mathrm{2}}\right)×−\mathrm{1}=−{sinx} \\ $$