1. **State the problem:** Differentiate the function $f(x) = \sin x$ with respect to $x$. 2. **Formula used:** The derivative of $\sin x$ with respect to $x$ is given by: $$\frac{d}{dx}(\sin x) = \cos x$$ 3. **Explanation:** The sine function is a basic trigonometric function, and its derivative is the cosine function. This is a fundamental rule in calculus. 4. **Example problem:** Differentiate $f(x) = \sin x$. 5. **Solution:** Using the formula, $$f'(x) = \cos x$$ 6. **Interpretation:** This means the rate of change of $\sin x$ at any point $x$ is equal to $\cos x$ at that point. **Final answer:** $$\frac{d}{dx}(\sin x) = \cos x$$