use the product rule
f(x)=color(red)(u)color(blue)(v)=>f'(x)=color(blue)(v)color(red)(u')+color(red)(u)color(blue)(v')
f(x)=color(red)(5secx)color(blue)(tanx)
color(red)(u=5secx=>u'=5secxtanx
color(blue)(v=tanx=>v'=sec^2x
:.f'(x)=color(blue)(tanx)color(red)(5secxtanx)+color(red)(5secx)color(blue)(sec^2x
f'(x)=5secxtan^2x+5sec^3x
f'(x)=5secx(tan^2x+sec^2x)
f'(x)=5secx(tan^2x+1+tan^2x))
f'(x)=5secx(2tan^2x+1)=10secxtan^2x+5secx