Differentiate the following w.r.t.x :
`2sqrt(cot(x^(2)))`

Answer

To differentiate the function \( y = 2\sqrt{\cot(x^2)} \) with respect to \( x \), we will use the chain rule. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
The function can be expressed as:
\[ y = 2\sqrt{u} \]
where \( u = \cot(x^2) \).

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( u \) is:
\[ \frac{dy}{du} = 2 \cdot \frac{1}{2\sqrt{u}} = \frac{1}{\sqrt{u}} \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( u = \cot(x^2) \) with respect to \( x \). The derivative of \( \cot(v) \) is \( -\csc^2(v) \cdot \frac{dv}{dx} \). Here, \( v = x^2 \), so:
\[ \frac{du}{dx} = -\csc^2(x^2) \cdot \frac{d(x^2)}{dx} = -\csc^2(x^2) \cdot 2x \]

### Step 4: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\cot(x^2)}} \cdot (-\csc^2(x^2) \cdot 2x) \]

### Step 5: Simplify the expression
Thus, we have:
\[ \frac{dy}{dx} = -\frac{2x \csc^2(x^2)}{\sqrt{\cot(x^2)}} \]

### Final Answer
The derivative of \( y = 2\sqrt{\cot(x^2)} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{2x \csc^2(x^2)}{\sqrt{\cot(x^2)}} \]

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