The function y = sec^2(2x) can be rewritten as y = sec(2x)^2 or y = g(x)^2 which should clue us in as a good candidate for the power rule.
The power rule: dy/dx = n* g(x)^(n-1) * d/dx(g(x))
where g(x) = sec(2x) and n=2 in our example.
Plugging these values into the power rule gives us
dy/dx = 2 * sec(2x) ^ 1 *d/dx(g(x))
Our only unknown remains d/dx(g(x)).
To find the derivative of g(x) = sec(2x), we need to use the chain rule because the inner part of g(x) is actually another function of x. In other words, g(x) = sec(h(x)).
The chain rule: g(h(x))' = g'(h(x)) * h'(x) where
g(x) = sec(h(x)) and
h(x) = 2x
g'(h(x)) = sec(h(x))tan(h(x))
h'(x) = 2
Let's use all of these values in the chain rule formula:
d/dx(g(x)) = d/dx(g(h(x))) = sec(2x)tan(x) * 2 = 2sec(2x)tan(x)
Now we can finally plug back this result into the power rule.
dy/dx = 2 * sec(2x) ^ 1 * d/dx(g(x))
dy/dx = 2sec(2x) * 2sec(2x)tan(x) = 4sec^2(2x)tan(2x)
