If you already know how to find the derivative of sec(2x), you don't need to change sec^2(2x) into 1/cos^2(2x).

I will go about this assuming you know how to take the derivative of secant.

f(x) = tan(2x)

The derivative of f(x) involves the chain rule (we need to use it twice):

f'(x) = sec^2(2x)*2 <-- the derivative of tan(2x) times the derivative of 2x.

f'(x) = 2sec^2(2x)

The derivative of f'(x) involves the chain rule (we need to use it three times):

f''(x) = 2*2sec(2x)*sec(2x)tan(2x)*2 <-- 2 times the derivatives of sec^2(2x) times the derivative of sec(2x) times the derivative of 2x.

f''(x) = 8sec^2(2x)tan(2x)