Question

How do you differentiate `f(theta)=sectheta/(1+sectheta)`?

Answer 1

`f'(theta)=(secthetatantheta)/(1+sectheta)^2`

Explanation:

using the quotient rule

`for y=u/v => (dy)/(dx)=(v(du)/(dx)-u(dv)/(dx))/v^2`

`f(theta)=sectheta/(1+sectheta)`

`u=sectheta=>(du)/(d(theta))=secthetatantheta`

`v=1+sectheta=>(dv)/(d(theta))=secthetatantheta`

`f'(theta)=((1+sectheta)(secthetatantheta)-sectheta(secthetatantheta))/(1+sectheta)^2`

`f'(theta)=(secthetatantheta+sec^2thetatantheta-sec^2thetatantheta)/(1+sectheta)^2`

`f'(theta)=(secthetatantheta+cancel(sec^2thetatantheta-sec^2thetatantheta))/(1+sectheta)^2`

`f'(theta)=(secthetatantheta)/(1+sectheta)^2`