1. Problem: Compute $\frac{d}{dx} (3 \sin x - 4 \cos x)$.\n\n2. Formula: The derivative of $\sin x$ is $\cos x$, and the derivative of $\cos x$ is $-\sin x$.\n\n3. Apply the derivative term-by-term:\n$$\frac{d}{dx} (3 \sin x) = 3 \cos x$$\n$$\frac{d}{dx} (-4 \cos x) = -4 (-\sin x) = 4 \sin x$$\n\n4. Combine the results:\n$$3 \cos x + 4 \sin x$$\n\n5. Explanation: We used the linearity of differentiation and the basic derivatives of sine and cosine functions.\n\nFinal answer: $3 \cos x + 4 \sin x$ (Option A).