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

Options

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

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = \cot^3(2x) \]
Here, we have an outer function \( u^3 \) where \( u = \cot(2x) \).

2. **Differentiate the outer function**:
Using the chain rule, the derivative of \( u^3 \) with respect to \( u \) is:
\[ \frac{dy}{du} = 3u^2 \]

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

4. **Apply the chain rule**:
Now, we can combine these results using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 3(\cot(2x))^2 \cdot (-2\csc^2(2x)) \]

5. **Simplify the expression**:
\[ \frac{dy}{dx} = -6(\cot(2x))^2 \csc^2(2x) \]

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