Obtain the differential coefficient of the following:
`log(cosx)`

Options

A. `tanx`
B. `-tanx`
C. `-cosx`
D. `-sinx`

Explanation

To find the differential coefficient of \( \log(\cos x) \), we will follow these steps:

### Step 1: Define the function
Let \( y = \log(\cos x) \).

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

### Step 3: Differentiate using the chain rule
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot \frac{d}{dx}(\cos x) \]

### Step 4: Differentiate \( \cos x \)
Now, we need to differentiate \( \cos x \):
\[ \frac{d}{dx}(\cos x) = -\sin x \]

### Step 5: Substitute back into the derivative
Substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) \]

### Step 6: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -\frac{\sin x}{\cos x} \]
which can be rewritten as:
\[ \frac{dy}{dx} = -\tan x \]

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

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