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Just use the chain rule:
(sec²x)' = 2secx.secxtanx = 2sec²xtanx
Basically, say y=sec²x, then let t=secx and y=t^2. Now:
dt/dx = secxtanx
dy/dt = 2t
dy/dx = (dy/dt)(dt/dx) = 2t.secxtanx = 2secx.secxtanx = 2sec²xtanx
To generalize, let's say you have y=u^n, where u is a function of x. Then:
dy/dx = n.(du/dx).u^(n-1)
(sec²x)' = 2secx.secxtanx = 2sec²xtanx
Basically, say y=sec²x, then let t=secx and y=t^2. Now:
dt/dx = secxtanx
dy/dt = 2t
dy/dx = (dy/dt)(dt/dx) = 2t.secxtanx = 2secx.secxtanx = 2sec²xtanx
To generalize, let's say you have y=u^n, where u is a function of x. Then:
dy/dx = n.(du/dx).u^(n-1)
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