To find the derivative of the function (1/8cos(x) - 3tan(x)), we need to use the chain rule and the derivative rules for cosine and tangent functions.

Let's break down the original function into two separate functions: f(x) = 1/8cos(x) and g(x) = 3tan(x).

The derivative of f(x) = 1/8cos(x) with respect to x is:
f'(x) = -1/8sin(x)

The derivative of g(x) = 3tan(x) with respect to x is:
g'(x) = 3sec^2(x)

Now, we apply the chain rule to find the derivative of the original function:

(1/8cos(x) - 3tan(x))' = f'(x) - g'(x)
= (-1/8sin(x)) - (3sec^2(x))
= -1/8sin(x) - 3sec^2(x)

Therefore, the derivative of the function (1/8cos(x) - 3tan(x)) is -1/8sin(x) - 3sec^2(x).