1. **State the problem:** We need to find the derivative of the function $$f(x) = \cos\left(\frac{3x}{2}\right)$$. 2. **Recall the formula:** The derivative of $$\cos(u)$$ with respect to $$x$$ is $$-\sin(u) \cdot \frac{du}{dx}$$, where $$u$$ is a function of $$x$$. 3. **Identify the inner function:** Here, $$u = \frac{3x}{2}$$. 4. **Compute the derivative of the inner function:** $$\frac{du}{dx} = \frac{3}{2}$$. 5. **Apply the chain rule:** $$ \frac{d}{dx} \cos\left(\frac{3x}{2}\right) = -\sin\left(\frac{3x}{2}\right) \cdot \frac{3}{2} = -\frac{3}{2} \sin\left(\frac{3x}{2}\right) $$ 6. **Final answer:** $$ \boxed{\frac{d}{dx} \cos\left(\frac{3x}{2}\right) = -\frac{3}{2} \sin\left(\frac{3x}{2}\right)} $$