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

Answer

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

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

### Step 2: Differentiate using the chain rule
We will apply the chain rule for differentiation. The derivative of \( \log(u) \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \]
where \( u = \sin x \).

### Step 3: Find \( \frac{du}{dx} \)
Now, we need to find the derivative of \( u = \sin x \):
\[ \frac{du}{dx} = \cos x \]

### Step 4: Substitute back into the derivative
Now we can substitute \( u \) and \( \frac{du}{dx} \) back into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{\sin x} \cdot \cos x \]

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

### Step 6: Recognize the trigonometric identity
The expression \( \frac{\cos x}{\sin x} \) is known as the cotangent function:
\[ \frac{dy}{dx} = \cot x \]

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

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