Differentiate the following w.r.t. x :
`sin(logx),xgt0`

Answer

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

### Step 1: Identify the function
Let \( y = \sin(\log x) \).

### Step 2: Differentiate using the chain rule
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by \( f'(g(x)) \cdot g'(x) \).

In our case:
- Let \( u = \log x \) (the inner function).
- Then \( y = \sin(u) \) (the outer function).

### Step 3: Differentiate the outer function
The derivative of \( \sin(u) \) with respect to \( u \) is:
\[ \frac{dy}{du} = \cos(u) \]

### Step 4: Differentiate the inner function
Next, we differentiate \( u = \log x \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{1}{x} \]

### Step 5: Apply the chain rule
Now, we apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(\log x) \cdot \frac{1}{x} \]

### Step 6: Write the final answer
Thus, the derivative of \( y = \sin(\log x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{\cos(\log x)}{x} \]

### Summary of Steps:
1. Identify the function and set \( y = \sin(\log x) \).
2. Use the chain rule to differentiate.
3. Differentiate the outer function \( \sin(u) \) to get \( \cos(u) \).
4. Differentiate the inner function \( \log x \) to get \( \frac{1}{x} \).
5. Combine the results using the chain rule.
6. Write the final answer.