Chain rule with tangent

Identify the outer function $f(u) = \tan(u)$ and inner function $u = g(x) = 3x - 2$. Since $f'(u) = \sec^2(u)$ and $u'(x) = 3$, the chain rule gives $g'(x) = \sec^2(3x-2)\cdot 3 = 3\sec^2(3x-2)$.

Now substitute $x = \pi/12$: the inner expression becomes $3(\pi/12) - 2 = \pi/4 - 2$. So $g'(\pi/12) = 3\sec^2(\pi/4 - 2)$.

To evaluate numerically: $\pi/4 - 2 \approx 0.7854 - 2 = -1.2146$. Then $\cos(-1.2146) \approx 0.3624$, so $\sec(-1.2146) \approx 2.76$ and $\sec^2(-1.2146) \approx 7.62$. Therefore, $g'(\pi/12) \approx 3 \times 7.62 \approx 22.86$.

The outer function is the tangent function. In g(x) = tan(3x - 2), the tangent function acts on the inner expression 3x - 2, so applying the chain rule requires differentiating tangent first, then multiplying by the derivative of the inner piece.