derivative of `cot ^2 x`

Answer

To find the derivative of \( \cot^2 x \), we can use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We are given the function \( y = \cot^2 x \).

### Step 2: Apply the chain rule
The derivative of \( y = u^2 \) where \( u = \cot x \) can be found using the chain rule:
\[ \frac{dy}{dx} = 2u \frac{du}{dx} \]
In our case, \( u = \cot x \).

### Step 3: Find the derivative of \( u \)
Now we need to find \( \frac{du}{dx} \) where \( u = \cot x \). The derivative of \( \cot x \) is:
\[ \frac{du}{dx} = -\csc^2 x \]

### Step 4: Substitute back into the chain rule
Now we substitute \( u \) and \( \frac{du}{dx} \) back into the equation from Step 2:
\[ \frac{dy}{dx} = 2(\cot x)(-\csc^2 x) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -2 \cot x \csc^2 x \]

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

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