Question Number 195136 by mathlove last updated on 25/Jul/23
$${f}\left({x}\right)={arctan}\left({sinx}\right) \\ $$$${and}\:\:{cosa}=\frac{\mathrm{2}}{\mathrm{3}}\:\:\:\:\:\:\:\:{faind}\:\:\:{f}^{'} \left({a}\right)=? \\ $$
Answered by som(math1967) last updated on 25/Jul/23
$$\:\boldsymbol{{f}}\:'\left({x}\right)=\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{{x}}}×\boldsymbol{{cosx}} \\ $$$$\boldsymbol{{f}}\:'\left(\boldsymbol{\alpha}\right)=\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{sin}}^{\mathrm{2}} \boldsymbol{\alpha}}×\boldsymbol{{cos}\alpha} \\ $$$$\:=\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{5}}{\mathrm{9}}}×\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$=\frac{\mathrm{9}}{\mathrm{14}}×\frac{\mathrm{2}}{\mathrm{3}}=\frac{\mathrm{3}}{\mathrm{7}} \\ $$