$$ \cot^2\left(\text{Senx}\right)^{\prime} $$

The given expression is:

\[
\left( \cot^2(\sin x) \right)^{\prime}
\]

To differentiate this, we apply the chain rule. The function involves both the square of a trigonometric function and a composite function inside it.

### Step-by-Step Differentiation

1. **Differentiate the outer function**:  
   The outer function is $ \cot^2(\sin x) $. Using the power rule and the chain rule, we differentiate $ u^2 $ with respect to $ u $, where $ u = \cot(\sin x) $. This gives:
   \[
   2 \cdot \cot(\sin x) \cdot \frac{d}{dx}[\cot(\sin x)]
   \]

2. **Differentiate $ \cot(\sin x) $**:  
   Now, we differentiate $ \cot(\sin x) $ using the chain rule. The derivative of $ \cot u $ is $ -\csc^2(u) $, so:
   \[
   \frac{d}{dx}[\cot(\sin x)] = -\csc^2(\sin x) \cdot \cos(x)
   \]

3. **Combine the results**:  
   Now, substitute this back into the original expression:
   \[
   2 \cdot \cot(\sin x) \cdot \left( -\csc^2(\sin x) \cdot \cos(x) \right)
   \]
   Simplifying:
   \[
   -2 \cot(\sin x) \cdot \csc^2(\sin x) \cdot \cos(x)
   \]

### Final Result:
\[
\boxed{-2 \cot(\sin x) \csc^2(\sin x) \cos(x)}
\]

Would you like a further breakdown of any steps?

Here are five questions related to this topic:
1. How do you differentiate composite trigonometric functions?
2. What is the chain rule and how does it apply to this problem?
3. Can you explain the derivative of $ \cot(x) $?
4. How would the differentiation change if the function was $ \cot(\cos(x)) $?
5. How is $ \csc(x) $ related to $ \sin(x) $?

**Tip**: When dealing with trigonometric functions, always apply the chain rule carefully, especially when multiple functions are nested inside one another.

Learn how to differentiate the expression $ \cot^2(\sin x) $ step-by-step using the chain rule. This solution covers trigonometric differentiation, the chain rule, and handling composite functions. Suitable for students in Grade 11-12 and early college calculus courses.

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