Derivative of `2sqrt(cot(x^(2))) ` with respect to x is

Options

A. `(x " cosec"^(2)(x^(2)))/(2sqrt(cot (x^(2))))`
B. `(-2x " cosec"^(2)(x^(2)))/(sqrt(cot (x^(2))))`
C. `(-x " cosec"^(2)(x^(2)))/(sqrt(cot (x^(2))))`
D. None of these

Explanation

To find the derivative of the function \( y = 2\sqrt{\cot(x^2)} \) with respect to \( x \), we will use the chain rule and the properties of derivatives. Let's go through the steps systematically.

### Step 1: Rewrite the function
First, we can rewrite the function for clarity:
\[ y = 2\sqrt{\cot(x^2)} = 2(\cot(x^2))^{1/2} \]

### Step 2: Differentiate using the chain rule
To differentiate \( y \), we will apply the chain rule. The derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 2 \cdot \frac{1}{2}(\cot(x^2))^{-1/2} \cdot \frac{d}{dx}[\cot(x^2)] \]
This simplifies to:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\cot(x^2)}} \cdot \frac{d}{dx}[\cot(x^2)] \]

### Step 3: Differentiate \( \cot(x^2) \)
Now we need to find \( \frac{d}{dx}[\cot(x^2)] \). The derivative of \( \cot(u) \) is \( -\csc^2(u) \cdot \frac{du}{dx} \). Here, \( u = x^2 \), so:
\[ \frac{d}{dx}[\cot(x^2)] = -\csc^2(x^2) \cdot \frac{d}{dx}[x^2] = -\csc^2(x^2) \cdot 2x \]

### Step 4: Substitute back into the derivative
Now we substitute this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\sqrt{\cot(x^2)}} \cdot (-\csc^2(x^2) \cdot 2x) \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{2x \csc^2(x^2)}{\sqrt{\cot(x^2)}} \]

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