To solve this question, we begin by determining the derivatives of the functions f(x) = tan(2x) and g(x) = sec(2x).
Firstly, let's find the derivative of f(x) = tan(2x). By applying the chain rule of differentiation, which states that the derivative of a composite function is the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function, we have:
f'(x) = (sec^2(2x)) * 2 = 2*sec^2(2x)
Next, we find the derivative of g(x) = sec(2x). The derivative of sec(x) is sec(x)tan(x), so by applying the chain rule again we get:
g'(x) = 2*sec(2x)*tan(2x)
So, the derivative of f(x) is 2*sec^2(2x), and the derivative of g(x) is 2*sec(2x)*tan(2x).
Now, let's find the difference between these two derivatives, i.e., f'(x) - g'(x):
We have:
Difference = f'(x) - g'(x) = 2*sec^2(2x) - 2*sec(2x)*tan(2x)
This expression cannot be simplified further without explicit values for x. Therefore, no particular pattern or relationship can be inferred from this difference without additional information or context.
In conclusion, the derivatives of the functions f(x) and g(x) are 2*sec^2(2x) and 2*sec(2x)*tan(2x) respectively, while the difference of the derivatives is 2*sec^2(2x) - 2*sec(2x)*tan(2x). Without further information or specific values of x, we cannot conclude anything more about the relationship between the derivatives.