derivative of log cos x

Answer

To find the derivative of the function \( f(x) = \log(\cos x) \), we will use the chain rule and the properties of logarithmic differentiation. Here is a step-by-step solution:

### Step 1: Identify the function
We have the function:
\[ f(x) = \log(\cos x) \]

### Step 2: Apply the chain rule
To differentiate \( f(x) \), we apply the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = \cos x \).

### Step 3: Differentiate the outer function
The derivative of \( \log(\cos x) \) with respect to \( \cos x \) is:
\[ \frac{d}{d(\cos x)} \log(\cos x) = \frac{1}{\cos x} \]

### Step 4: Differentiate the inner function
Next, we need to find the derivative of the inner function \( \cos x \):
\[ \frac{d}{dx}(\cos x) = -\sin x \]

### Step 5: Combine the results using the chain rule
Now, we combine the results from Steps 3 and 4:
\[ f'(x) = \frac{1}{\cos x} \cdot (-\sin x) \]

### Step 6: Simplify the expression
This simplifies to:
\[ f'(x) = -\frac{\sin x}{\cos x} \]
Using the identity \( \tan x = \frac{\sin x}{\cos x} \), we can rewrite this as:
\[ f'(x) = -\tan x \]

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

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