cosec`sqrt(x)`

Answer

To differentiate the function \( y = \csc(\sqrt{x}) \), we will use the chain rule and the derivatives of the cosecant and square root functions.

### Step-by-Step Solution:

1. **Identify the Function**:
We have \( y = \csc(\sqrt{x}) \).

2. **Differentiate Using the Chain Rule**:
According to the chain rule, if \( y = \csc(u) \) where \( u = \sqrt{x} \), then:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

3. **Find \( \frac{dy}{du} \)**:
The derivative of \( \csc(u) \) is:
\[ \frac{dy}{du} = -\csc(u) \cot(u) \]
So, substituting \( u = \sqrt{x} \):
\[ \frac{dy}{du} = -\csc(\sqrt{x}) \cot(\sqrt{x}) \]

4. **Find \( \frac{du}{dx} \)**:
Now, we need to differentiate \( u = \sqrt{x} \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^{1/2}) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

5. **Combine the Results**:
Now, substitute \( \frac{dy}{du} \) and \( \frac{du}{dx} \) back into the chain rule:
\[ \frac{dy}{dx} = -\csc(\sqrt{x}) \cot(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

6. **Final Result**:
Therefore, the derivative of \( y = \csc(\sqrt{x}) \) is:
\[ \frac{dy}{dx} = -\frac{\csc(\sqrt{x}) \cot(\sqrt{x})}{2\sqrt{x}} \]