The function f(x) = sec (4x)

f'(x) = `4*sec 4x*tan 4x`

f''(x) = `16*sec 4x*tan 4x*tan 4x + 16*sec 4x*sec^2 4x`

=> `16*sec 4x*tan^2 4x + 16*sec^3 4x`

The required second derivative of `f(x) = sec 4x` is `f''(x) = 16*sec 4x*tan^2 4x + 16*sec^3 4x`

`f(x) = sec(4x)`

`f'(x) = 4 * sec(4x) * tan(4x)`

Explanation,

take 4x = u

then 4 dx = du, or du/dx = 4

f(x) = y = sec(u)

dy/du = sec(u)*tan(u)

dy/dx = (dy/du) * (du/dx)

dy/dx = sec(u)*tan(u)*4

therefore,

dy/dx = 4*sec(4x)*tan(4x)