Question

How do you differentiate `f(x) =cosx*sec^2(x)`?

Answer 1

`d/(d x) f(x)=sec^2 x`

Explanation:

`d/(d x) f(x)=-sin x *sec^2 x+ 2secx*sec x*tan x*cos x`
`d/(d x) f(x)=-sin x *sec^2 x+ 2sec^2x*tan x*cos x`
`d/(d x) f(x)=sec^2(-sin x+tan x*cos x)" , "tan x=(sin x)/(cos x)`
`d/(d x) f(x)=sec^2 x(-sin x+(sin x)/cancel(cos x)*cancel(cos x))`
`d/(d x) f(x)=sec^2 x(cancel(-sin x)+cancel(sin x))`
`d/(d x) f(x)=sec^2 x`

Answer 2

`f'(x) = secxtanx`. Here are two ways to get this answer.

Explanation:

Rewrite first

`f(x)=cosxsec^2x`

`= cosx 1/(cosx)^2`

`= 1/cosx`

`= secx`.

So,

`f'(x) = secxtanx`.

Use product rule first
`f(x)=cosxsec^2x`

`f'(x) = (-sinx)(sec^2x)+(cosx)(2secx*secxtanx)`

`= -sinxsec^2x+2sec^2x [tanx cosx]`

`= -sinxsec^2x+2sec^2x [sinx]`

`= sec^2xsinx`

`= secx tanx`