log cos x

Answer

To find the derivative of the function \( y = \log(\cos x) \) with respect to \( x \), we will use the chain rule and the derivatives of logarithmic and trigonometric functions.

### Step-by-Step Solution:

1. **Identify the function**: We have \( y = \log(\cos x) \).

2. **Apply the chain rule**: The derivative of \( \log(u) \) with respect to \( x \) is given by \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \cos x \).

3. **Differentiate \( \log(\cos x) \)**:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}(\cos x) \]

4. **Differentiate \( \cos x \)**: The derivative of \( \cos x \) is \( -\sin x \).
\[ \frac{d}{dx}(\cos x) = -\sin x \]

5. **Substitute back into the derivative**:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) \]

6. **Simplify the expression**:
\[ \frac{dy}{dx} = -\frac{\sin x}{\cos x} \]

7. **Recognize the trigonometric identity**: The expression \( -\frac{\sin x}{\cos x} \) can be rewritten as:
\[ \frac{dy}{dx} = -\tan x \]

### Final Result:
Thus, the derivative of \( y = \log(\cos x) \) is:
\[ \frac{dy}{dx} = -\tan x \]