log {log (cos x}

Answer

To differentiate the function \( y = \log(\log(\cos x)) \) with respect to \( x \), we will apply the chain rule multiple times. Here are the steps:

### Step 1: Identify the outer and inner functions
The function can be broken down as follows:
- Outer function: \( u = \log(v) \) where \( v = \log(\cos x) \)
- Inner function: \( v = \log(\cos x) \)

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{1}{v} \cdot \frac{dv}{dx} \]
Substituting \( v \):
\[ \frac{dy}{dx} = \frac{1}{\log(\cos x)} \cdot \frac{dv}{dx} \]

### Step 3: Differentiate the inner function
Now we need to differentiate \( v = \log(\cos x) \):
Using the chain rule again:
\[ \frac{dv}{dx} = \frac{1}{\cos x} \cdot \frac{d(\cos x)}{dx} \]
The derivative of \( \cos x \) is \( -\sin x \):
\[ \frac{dv}{dx} = \frac{1}{\cos x} \cdot (-\sin x) = -\frac{\sin x}{\cos x} = -\tan x \]

### Step 4: Substitute back into the derivative
Now, substituting \( \frac{dv}{dx} \) back into the expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\log(\cos x)} \cdot (-\tan x) \]
Thus, we have:
\[ \frac{dy}{dx} = -\frac{\tan x}{\log(\cos x)} \]

### Final Answer
The derivative of \( y = \log(\log(\cos x)) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{\tan x}{\log(\cos x)} \]