`d/(dx)(xcosecx)=……………………` .

Answer

To find the derivative of the function \( y = x \cdot \csc x \), we will use the product rule of differentiation. The product rule states that if you have two functions multiplied together, \( u \) and \( v \), then the derivative \( \frac{d}{dx}(uv) \) is given by:

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

In our case, let:
- \( u = x \)
- \( v = \csc x \)

### Step 1: Differentiate \( u \) and \( v \)

1. Differentiate \( u = x \):
\[ \frac{du}{dx} = 1 \]

2. Differentiate \( v = \csc x \):
The derivative of \( \csc x \) is:
\[ \frac{dv}{dx} = -\csc x \cot x \]

### Step 2: Apply the Product Rule

Now, applying the product rule:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = x \cdot (-\csc x \cot x) + \csc x \cdot 1 \]
This simplifies to:
\[ \frac{dy}{dx} = -x \csc x \cot x + \csc x \]

### Step 3: Factor out \( \csc x \)

We can factor out \( \csc x \):
\[ \frac{dy}{dx} = \csc x (-x \cot x + 1) \]

### Final Answer

Thus, the derivative of \( y = x \cdot \csc x \) is:
\[ \frac{dy}{dx} = \csc x (1 - x \cot x) \]

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