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 are the steps:

### Step 1: Apply the Chain Rule
We start with the function:
\[ y = \log(\cos(5x)) \]
Using the chain rule, the derivative of \( \log(u) \) where \( u = \cos(5x) \) is given by:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \]
So, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos(5x)} \cdot \frac{d}{dx}(\cos(5x)) \]

### Step 2: Differentiate \( \cos(5x) \)
Next, 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 3: Substitute Back into the Derivative
Now, we substitute \( \frac{d}{dx}(\cos(5x)) \) back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\cos(5x)} \cdot (-5\sin(5x)) \]

### Step 4: Simplify the Expression
This simplifies to:
\[ \frac{dy}{dx} = -5 \cdot \frac{\sin(5x)}{\cos(5x)} \]
Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we can further simplify:
\[ \frac{dy}{dx} = -5 \tan(5x) \]

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

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