Derivative Cosine 6Aabf3

1. **Problem Statement:** Verify that the derivative of $-\cos x$ is $\sin x$. 2. **Formula and Theory:** The derivative of $\cos x$ with respect to $x$ is $-\sin x$. Using the constant multiple rule, the derivative of $-\cos x$ is $-1$ times the derivative of $\cos x$. 3. **Step-by-step Solution:** - Start with the function: $f(x) = -\cos x$ - Apply the derivative operator: $f'(x) = \frac{d}{dx}(-\cos x) = - \frac{d}{dx}(\cos x)$ - Use the known derivative: $\frac{d}{dx}(\cos x) = -\sin x$ - Substitute: $f'(x) = -(-\sin x) = \sin x$ 4. **Explanation:** The derivative of $\cos x$ is $-\sin x$ because the cosine function decreases at a rate proportional to the sine function. Multiplying by $-1$ changes the sign, so the derivative of $-\cos x$ becomes $\sin x$. **Final answer:** $$\frac{d}{dx}(-\cos x) = \sin x$$