A

To find \( \frac{d\alpha}{d\beta} \) given \( \alpha = \sec(3\beta) \), we need to apply the chain rule of differentiation.
Step 1: Differentiate \( \alpha \):
\( \frac{d\alpha}{d\beta} = \frac{d}{d\beta}[\sec(3\beta)] \)
Step 2: Recall that the derivative of \( \sec(x) \) is \( \sec(x)\tan(x) \). Thus:
\( \frac{d}{d\beta}[\sec(3\beta)] = \sec(3\beta)\tan(3\beta) \cdot \frac{d}{d\beta}[3\beta] \)
Step 3: Apply the derivative of the inner function:
\( \frac{d}{d\beta}[3\beta] = 3 \).
Step 4: Substitute this back:
\( \frac{d\alpha}{d\beta} = \sec(3\beta)\tan(3\beta) \cdot 3 = 3\sec(3\beta)\tan(3\beta) \).
Therefore, the result is \( \frac{d\alpha}{d\beta} = 3 \sec(3\beta) \tan(3\beta) \).