The derivative of f when?

Question

F(x)= (1+sinx) / (cosx)

Is it (– 1) / ( 1+cosx)

Answer 1

Using quotient rule:-
f `(x) = [ cosx.cosx + (1 + sinx).sinx ] / cos² x
f `(x) = (cos² x + sin x + sin² x) / cos² x
f `(x) = (1 + sin x) / (1 - sin²x)
f `(x) = (1 + sin x) / [ (1 - sin x).(1 + sin x) ]
f `(x) = 1 / (1 - sin x)

Answer 2

There is an easy way to do this without using the quotient rule (and the person's answer above me is wrong):

F(x)= (1+sinx) / (cosx)
= (1/cosx) + (sinx/cosx)
= (secx) + (tanx)

F'(x) = secxtanx + (secx)^2
= (1/cosx)(sinx/cosx) + (1/cosx)(1/cosx)
= (sinx/[(cosx)^2]) + (1/[(cosx)^2])
= (sinx + 1) / [(cosx)^2]
= (sinx + 1) / (1 - (sinx)^2) [trig identity]
= (sinx + 1) / [(1-sinx)(1+sinx)]

= 1 / (1 - sinx)

Answer 3

use the quotient rule: d(u/v) = ( vdu - udv ) / v^2

F(x)= (1+sinx) / (cosx)
F'(x) = ( (cosx)(sinx) - (1+sinx)(-cosx) ) / (cos x)^2
= -1 / (cosx)^2

Answer 4

f, or F?