first, the the derivative of y=lnA is y' = A' / A
let y=(tanx)/(1+secx), implies lny=ln [(tanx)/(1+secx)]
(lny)' =(ln [(tanx)/(1+secx)])' so y' /y = [(tanx)/(1+secx)])'/ (tanx)/(1+secx)
(tanx)/(1+secx)' =[(tanx)' (1+secx) - (tanx)(1+secx)'] / (1+secx)²
(tanx)' = 1 + tan²x and (1+secx)']= secx tanx (check lesson)
so
(tanx)/(1+secx)' = (1 + tan²x) (1+secx) - ( tan²xsecx ) / (1+secx)²
= (1 + secx +tan²x) / (1+secx)²
y' /y = [(tanx)/(1+secx)])'/ (tanx)/(1+secx)=
y' /y =[(1 + secx +tan²x) / (1+secx)² ] / (tanx)/(1+secx)
y' /y =(1 + secx +tan²x) (1+secx) / (tanx) (1+secx)²
so y' = y. [ (1 + secx +tan²x) (1+secx) / (tanx) (1+secx)² ]
but y=(tanx)/(1+secx),
y' = (tanx)/(1+secx) [ (1 + secx +tan²x) (1+secx) / (tanx) (1+secx)² ]
finally the derivative of (tanx)/(1+secx) is given by
y' = (1 + secx +tan²x)