derivative of cosec`sqrt(x)`

Answer

To find the derivative of the function \( y = \csc(\sqrt{x}) \), we will apply the chain rule and the derivative of the cosecant function.

### Step-by-Step Solution:

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

2. **Recall the Derivative of Cosecant**:
The derivative of \( \csc(u) \) with respect to \( u \) is:
\[ \frac{d}{du} \csc(u) = -\csc(u) \cot(u) \]

3. **Apply the Chain Rule**:
Since \( u = \sqrt{x} \), we need to use the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Thus,
\[ \frac{dy}{dx} = -\csc(\sqrt{x}) \cot(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]

4. **Differentiate \( \sqrt{x} \)**:
We know that:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]

5. **Substitute Back**:
Now, substituting \( \frac{d}{dx}(\sqrt{x}) \) back into our derivative expression:
\[ \frac{dy}{dx} = -\csc(\sqrt{x}) \cot(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

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