derivative of log {log (cos x}

Answer

To find the derivative of the function \( y = \log(\log(\cos x)) \), we will use the chain rule of differentiation. Here are the steps to solve the problem:

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

### Step 2: Differentiate the outer function
Using the derivative of the logarithmic function, we know that:
\[ \frac{du}{dv} = \frac{1}{v} \]
Thus, we have:
\[ \frac{du}{dv} = \frac{1}{\log(\cos x)} \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( v = \log(\cos x) \). Again, using the chain rule:
\[ \frac{dv}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}(\cos x) \]
Since the derivative of \( \cos x \) is \( -\sin x \), we have:
\[ \frac{dv}{dx} = \frac{-\sin x}{\cos x} = -\tan x \]

### Step 4: Apply the chain rule
Now we can apply the chain rule to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\log(\cos x)} \cdot (-\tan x) \]

### Step 5: Simplify the expression
Thus, the derivative of \( y = \log(\log(\cos x)) \) is:
\[ \frac{dy}{dx} = -\frac{\tan x}{\log(\cos x)} \]

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

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