Question Number 122434 by benjo_mathlover last updated on 17/Nov/20
$$\:{If}\:{f}\left({x}\right)\:=\:\int_{{a}} ^{\:\left(\underset{{a}} {\overset{{x}^{\mathrm{3}} } {\int}}\:\frac{{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} {t}}\:\right)} \left(\frac{{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} {t}}\right) \\ $$$${then}\:{f}\:'\left({x}\right)\:? \\ $$
Answered by liberty last updated on 17/Nov/20
$$\:\mathrm{f}\:'\left(\mathrm{x}\right)\:=\:\frac{\left(\mathrm{x}^{\mathrm{3}} \right)'}{\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\right).\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\underset{\mathrm{1}} {\overset{\mathrm{x}^{\mathrm{3}} } {\int}}\:\frac{\mathrm{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{t}}\right)\right)\:} \\ $$$$\:\mathrm{f}\:'\left(\mathrm{x}\right)\:=\:\frac{\mathrm{3x}^{\mathrm{2}} }{\left(\mathrm{1}+\mathrm{2sin}\:^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{3}} \right)\right).\left(\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \left(\underset{\mathrm{1}} {\overset{\mathrm{x}^{\mathrm{3}} } {\int}}\:\frac{\mathrm{dt}}{\mathrm{1}+\mathrm{sin}\:^{\mathrm{2}} \mathrm{t}}\right)\right)} \\ $$