Evaluate differentiation of y with respect to x.
`((2)/(sinx))`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \), where \( y = \frac{2}{\sin x} \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \frac{2}{\sin x} \]
For simplicity, we can rewrite this as:
\[ y = 2 \cdot \sin^{-1}(x) \]

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we will use the chain rule. We can express the derivative \( \frac{dy}{dx} \) as:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]
where we let \( t = \sin x \).

### Step 3: Differentiate \( y \) with respect to \( t \)
Now, we differentiate \( y \) with respect to \( t \):
\[ y = 2t^{-1} \]
Using the power rule, we differentiate:
\[ \frac{dy}{dt} = -2t^{-2} = -\frac{2}{t^2} \]

### Step 4: Differentiate \( t \) with respect to \( x \)
Next, we differentiate \( t = \sin x \) with respect to \( x \):
\[ \frac{dt}{dx} = \cos x \]

### Step 5: Combine the derivatives
Now we can combine the derivatives:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = -\frac{2}{t^2} \cdot \cos x \]
Substituting back \( t = \sin x \):
\[ \frac{dy}{dx} = -\frac{2}{\sin^2 x} \cdot \cos x \]

### Step 6: Final result
Thus, the final result for the differentiation of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{2 \cos x}{\sin^2 x} \]