Differentiate the following function w.r.t. x:
`sqrt (x) cosec (5x +7)`

Answer

To differentiate the function \( y = \sqrt{x} \cdot \csc(5x + 7) \) with respect to \( x \), we will use the product rule and the chain rule.

### Step-by-Step Solution:

1. **Identify the components of the function:**
Let \( u = \sqrt{x} \) and \( v = \csc(5x + 7) \).

2. **Differentiate \( u \):**
\[ u = x^{1/2} \implies \frac{du}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

3. **Differentiate \( v \):**
To differentiate \( v = \csc(5x + 7) \), we use the chain rule:
\[ \frac{dv}{dx} = -\csc(5x + 7) \cot(5x + 7) \cdot \frac{d}{dx}(5x + 7) \]
\[ \frac{d}{dx}(5x + 7) = 5 \implies \frac{dv}{dx} = -5 \csc(5x + 7) \cot(5x + 7) \]

4. **Apply the product rule:**
The product rule states that if \( y = uv \), then:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = \sqrt{x} \cdot \left(-5 \csc(5x + 7) \cot(5x + 7)\right) + \csc(5x + 7) \cdot \frac{1}{2\sqrt{x}} \]

5. **Combine the terms:**
\[ \frac{dy}{dx} = -5\sqrt{x} \csc(5x + 7) \cot(5x + 7) + \frac{1}{2\sqrt{x}} \csc(5x + 7) \]

6. **Factor out common terms:**
\[ \frac{dy}{dx} = \csc(5x + 7) \left(-5\sqrt{x} \cot(5x + 7) + \frac{1}{2\sqrt{x}}\right) \]

### Final Answer:
\[ \frac{dy}{dx} = \csc(5x + 7) \left(-5\sqrt{x} \cot(5x + 7) + \frac{1}{2\sqrt{x}}\right) \]