Question Number 146046 by mathdanisur last updated on 10/Jul/21
$${f}\left({x}\right)\:=\:{x}^{\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)} \:\Rightarrow\:{f}\:^{'} \left({x}\right)\:=\:? \\ $$
Answered by liberty last updated on 10/Jul/21
$$\:\mathrm{ln}\:{f}\left({x}\right)=\mathrm{sin}\:{x}.\:\mathrm{ln}\:\left({x}\right) \\ $$$$\Leftrightarrow\:\frac{{f}\:'\left({x}\right)}{{f}\left({x}\right)}=\:\mathrm{cos}\:{x}.\mathrm{ln}\:\left({x}\right)+\frac{\mathrm{sin}\:{x}}{{x}} \\ $$$$\Leftrightarrow\:{f}\:'\left({x}\right)={x}^{\mathrm{sin}\:{x}\:} \left(\mathrm{cos}\:{x}.\mathrm{ln}\:\left({x}\right)+\frac{\mathrm{sin}\:{x}}{{x}}\right) \\ $$
Commented by mathdanisur last updated on 10/Jul/21
$${thankyou}\:{Ser} \\ $$