1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = -4 \sin(x) + 9x.$$\n\n2. **Recall the derivative rules:**\n- The derivative of $\sin(x)$ is $\cos(x)$.\n- The derivative of $x$ is 1.\n- The derivative of a constant times a function is the constant times the derivative of the function.\n\n3. **Apply the derivative operator:**\n$$\frac{d}{dx}[-4 \sin(x) + 9x] = \frac{d}{dx}[-4 \sin(x)] + \frac{d}{dx}[9x].$$\n\n4. **Differentiate each term:**\n$$\frac{d}{dx}[-4 \sin(x)] = -4 \frac{d}{dx}[\sin(x)] = -4 \cos(x),$$\n$$\frac{d}{dx}[9x] = 9 \frac{d}{dx}[x] = 9 \cdot 1 = 9.$$\n\n5. **Combine the results:**\n$$\frac{d}{dx}[-4 \sin(x) + 9x] = -4 \cos(x) + 9.$$\n\n**Final answer:** $$\boxed{-4 \cos(x) + 9}.$$