Question Number 150186 by mathdanisur last updated on 10/Aug/21
$$\mathrm{If}\:\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{sin}^{\mathrm{4}} \left(\mathrm{3x}\right)\:\:\Rightarrow\:\:{f}\:^{'} \left(\frac{\pi}{\mathrm{12}}\right)\:=\:? \\ $$
Answered by puissant last updated on 10/Aug/21
$$\mathrm{f}'\left(\mathrm{x}\right)=\mathrm{4}×\mathrm{3cos}\left(\mathrm{3x}\right)\mathrm{sin}^{\mathrm{3}} \left(\mathrm{3x}\right) \\ $$$$\Rightarrow\:\mathrm{f}'\left(\mathrm{x}\right)=\mathrm{12cos}\left(\mathrm{3x}\right)\mathrm{sin}^{\mathrm{3}} \left(\mathrm{3x}\right) \\ $$$$\mathrm{f}'\left(\frac{\pi}{\mathrm{12}}\right)=\mathrm{12}×\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}×\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)^{\mathrm{3}} =\mathrm{12}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)^{\mathrm{4}} =\mathrm{12}\left(\frac{\mathrm{4}}{\mathrm{16}}\right) \\ $$$$\mathrm{f}'\left(\frac{\pi}{\mathrm{12}}\right)=\:\mathrm{3}... \\ $$
Commented by mathdanisur last updated on 10/Aug/21
$${thank}\:{you}\:\:{Ser} \\ $$