Differentiate the following w.r.t. x :
`sqrt(log(sin(x^(2)/3-1)))`

Answer

To differentiate the function \( y = \sqrt{\log(\sin(\frac{x^2}{3}) - 1)} \) with respect to \( x \), we will use the chain rule and the properties of derivatives. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
The function can be expressed as:
- Outer function: \( u = \sqrt{v} \) where \( v = \log(\sin(\frac{x^2}{3}) - 1) \)
- Inner function: \( v = \log(\sin(\frac{x^2}{3}) - 1) \)

### Step 2: Differentiate the outer function
Using the chain rule:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{v}} \cdot \frac{dv}{dx} \]

### Step 3: Differentiate the inner function
Now, we need to differentiate \( v = \log(\sin(\frac{x^2}{3}) - 1) \):
\[ \frac{dv}{dx} = \frac{1}{\sin(\frac{x^2}{3}) - 1} \cdot \frac{d}{dx}(\sin(\frac{x^2}{3}) - 1) \]
The derivative of \( \sin(\frac{x^2}{3}) \) using the chain rule is:
\[ \frac{d}{dx}(\sin(\frac{x^2}{3})) = \cos(\frac{x^2}{3}) \cdot \frac{d}{dx}(\frac{x^2}{3}) = \cos(\frac{x^2}{3}) \cdot \frac{2x}{3} \]
Thus,
\[ \frac{dv}{dx} = \frac{1}{\sin(\frac{x^2}{3}) - 1} \cdot \left(\cos(\frac{x^2}{3}) \cdot \frac{2x}{3}\right) \]

### Step 4: Substitute back into the derivative
Now substituting \( \frac{dv}{dx} \) back into the expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{\log(\sin(\frac{x^2}{3}) - 1)}} \cdot \frac{1}{\sin(\frac{x^2}{3}) - 1} \cdot \left(\cos(\frac{x^2}{3}) \cdot \frac{2x}{3}\right) \]

### Step 5: Simplify the expression
Combining everything, we have:
\[ \frac{dy}{dx} = \frac{x \cos(\frac{x^2}{3})}{3 \sqrt{\log(\sin(\frac{x^2}{3}) - 1) (\sin(\frac{x^2}{3}) - 1)}} \]

### Final Answer
Thus, the derivative of \( y = \sqrt{\log(\sin(\frac{x^2}{3}) - 1)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{x \cos(\frac{x^2}{3})}{3 \sqrt{\log(\sin(\frac{x^2}{3}) - 1) (\sin(\frac{x^2}{3}) - 1)}} \]