Differentiate the following w.r.t. x :
`log(cos5x)`

Answer

To differentiate the function \( y = \log(\cos(5x)) \) with respect to \( x \), we will use the chain rule and the properties of logarithmic differentiation. Here’s a step-by-step solution:

### Step 1: Write the function
Let:
\[ y = \log(\cos(5x)) \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \), where \( u = \cos(5x) \).

Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos(5x)} \cdot \frac{d}{dx}(\cos(5x)) \]

### Step 3: Differentiate \( \cos(5x) \)
Now, we need to differentiate \( \cos(5x) \). Using the chain rule again:
\[ \frac{d}{dx}(\cos(5x)) = -\sin(5x) \cdot \frac{d}{dx}(5x) = -\sin(5x) \cdot 5 = -5\sin(5x) \]

### Step 4: Substitute back into the derivative
Now, substitute \( \frac{d}{dx}(\cos(5x)) \) back into the derivative of \( y \):
\[ \frac{dy}{dx} = \frac{1}{\cos(5x)} \cdot (-5\sin(5x)) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -5 \cdot \frac{\sin(5x)}{\cos(5x)} = -5\tan(5x) \]

### Final Answer
Thus, the derivative of \( y = \log(\cos(5x)) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -5\tan(5x) \]

---