`cot ^2 x`

Answer

To differentiate the function \( \cot^2 x \), we will follow these steps:

### Step-by-Step Solution:

1. **Identify the function**: We have the function \( y = \cot^2 x \).

2. **Apply the power rule**: The derivative of \( y = u^n \) (where \( u = \cot x \) and \( n = 2 \)) is given by:
\[ \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \]
Here, \( n = 2 \) and \( u = \cot x \).

3. **Differentiate \( u = \cot x \)**: The derivative of \( \cot x \) is:
\[ \frac{du}{dx} = -\csc^2 x \]

4. **Substitute into the derivative formula**:
\[ \frac{dy}{dx} = 2 \cdot \cot^{2-1} x \cdot \frac{du}{dx} = 2 \cdot \cot x \cdot (-\csc^2 x) \]

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

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

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