The differential coefficient of the function `cot^3 2x` is:

Options

A. `-6 cot^2 2x cdot cosec^2 2x`
B. `-3 cot^2 2x cdot cosec^2 2x`
C. `-6 cot^2 2x cdot cosec 2x`
D. `-3 cot^2 2x cdot cosec 2x`

Answer

To find the differential coefficient of the function \( y = \cot^3(2x) \), we will use the chain rule and the derivative of the cotangent function. Here’s a step-by-step solution:

### Step 1: Identify the function
We start with the function:
\[ y = \cot^3(2x) \]

### Step 2: Apply the chain rule
To differentiate \( y \), we will use the chain rule. The derivative of \( u^n \) is \( n \cdot u^{n-1} \cdot \frac{du}{dx} \). Here, \( u = \cot(2x) \) and \( n = 3 \).

So, we have:
\[ \frac{dy}{dx} = 3 \cdot \cot^2(2x) \cdot \frac{d}{dx}(\cot(2x)) \]

### Step 3: Differentiate \( \cot(2x) \)
Next, we need to differentiate \( \cot(2x) \). The derivative of \( \cot(x) \) is \( -\csc^2(x) \). Therefore, using the chain rule again:
\[ \frac{d}{dx}(\cot(2x)) = -\csc^2(2x) \cdot \frac{d}{dx}(2x) = -\csc^2(2x) \cdot 2 \]

### Step 4: Substitute back into the derivative
Now, substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3 \cdot \cot^2(2x) \cdot (-2 \csc^2(2x)) \]
\[ \frac{dy}{dx} = -6 \cot^2(2x) \csc^2(2x) \]

### Final Result
Thus, the differential coefficient of the function \( \cot^3(2x) \) is:
\[ \frac{dy}{dx} = -6 \cot^2(2x) \csc^2(2x) \]