What is the exact derivative of secant x evaluated at x = π/6 radians?

To find the exact derivative of secant x evaluated at x = π/6 radians, we begin by recalling the derivative of the secant function. The derivative of secant x is given by:

[ \frac{d}{dx}(\sec x) = \sec x \tan x. ]

Next, we need to evaluate this derivative at x = π/6. First, we need to find the values of sec(π/6) and tan(π/6):

1. Secant function:

[ \sec(x) = \frac{1}{\cos(x)}. ]

The cosine of π/6 is ( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} ), so:

[ \sec(\frac{\pi}{6}) = \frac{1}{\cos(\frac{\pi}{6})} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}. ]

2. Tangent function:

[ \tan(x) = \frac{\sin(x)}{\cos(x)}. ]