Derivative of sec x

y = sec³ x means y = (sec x)³.

This is a composite function of
y = (whole)³ and (whole) = sec x.

So the derivative of the composite function is,
the derivative of the outer function (sec x)³, 3(sec x)²
times,
the derivative of the inner function sec x, sec x tan x.

(sec x)²⋅(sec x) = (sec x)³ = sec³ x

So the right side is 3 sec³ x tan x.

So 3 sec³ x tan x is the derivative of sec³ x.