Differentiate the following w.r.t.x :
`cosec(cotsqrtx)`

Answer

To differentiate the function \( y = \csc(\cot(\sqrt{x})) \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
The outer function is \( \csc(u) \) where \( u = \cot(\sqrt{x}) \).

### Step 2: Differentiate the outer function
The derivative of \( \csc(u) \) is given by:
\[ \frac{d}{du}(\csc(u)) = -\csc(u) \cot(u) \]
So, we have:
\[ \frac{dy}{du} = -\csc(\cot(\sqrt{x})) \cot(\cot(\sqrt{x})) \]

### Step 3: Differentiate the inner function
Now we need to differentiate the inner function \( u = \cot(\sqrt{x}) \). The derivative of \( \cot(v) \) is:
\[ \frac{d}{dv}(\cot(v)) = -\csc^2(v) \]
where \( v = \sqrt{x} \). Therefore, we need to apply the chain rule again:
\[ \frac{du}{dx} = -\csc^2(\sqrt{x}) \cdot \frac{d}{dx}(\sqrt{x}) \]

### Step 4: Differentiate \( \sqrt{x} \)
The derivative of \( \sqrt{x} \) is:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
Thus, we have:
\[ \frac{du}{dx} = -\csc^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]

### Step 5: Combine the derivatives using the chain rule
Now we can combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \left(-\csc(\cot(\sqrt{x})) \cot(\cot(\sqrt{x}))\right) \cdot \left(-\csc^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}\right) \]

### Step 6: Simplify the expression
The negatives cancel out:
\[ \frac{dy}{dx} = \csc(\cot(\sqrt{x})) \cot(\cot(\sqrt{x})) \cdot \csc^2(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \]
Thus, the final derivative is:
\[ \frac{dy}{dx} = \frac{\csc(\cot(\sqrt{x})) \cot(\cot(\sqrt{x})) \csc^2(\sqrt{x})}{2\sqrt{x}} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{\csc(\cot(\sqrt{x})) \cot(\cot(\sqrt{x})) \csc^2(\sqrt{x})}{2\sqrt{x}} \]

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