Calculate the derivative of the following functions.
(a) f (x) = cot2 x 

To find the derivative of the function \( f(x) = \cot^2{x} \), we need to apply the chain rule and the derivative of the cotangent function.

1. Start by rewriting \( f(x) = \cot^2{x} \) as:
   \[
   f(x) = \left( \cot{x} \right)^2
   \]

2. Apply the chain rule. If \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \). Here, \( g(u) = u^2 \) and \( h(x) = \cot{x} \).

3. First, differentiate the outer function \( g(u) = u^2 \) with respect to \( u \):
   \[
   \frac{d}{du} \left( u^2 \right) = 2u
   \]
   So, we get:
   \[
   \frac{d}{dx} \left( \cot^2{x} \right) = 2 \cot{x} \cdot \frac{d}{dx} \left( \cot{x} \right)
   \]

4. Now, differentiate the inner function \( \cot{x} \) with respect to \( x \):
   \[
   \frac{d}{dx} \left( \cot{x} \right) = -\csc^2{x}
   \]

5. Combining these results:
   \[
   \frac{d}{dx} \left( \cot^2{x} \right) = 2 \cot{x} \cdot \left( -\csc^2{x} \right)
   \]
   \[
   \frac{d}{dx} \left( \cot^2{x} \right) = -2 \cot{x} \csc^2{x}
   \]

Thus, the derivative of \( f(x) = \cot^2{x} \) is:
\[
f'(x) = -2 \cot{x} \csc^2{x}
\]

Would you like further details or any clarifications? 

Here are some related questions:
1. How do you apply the product rule to trigonometric functions?
2. What is the derivative of \( \csc^2{x} \)?
3. Can you use implicit differentiation for similar trigonometric problems?
4. How would the derivative change if \( f(x) = \tan^2{x} \)?
5. What happens when differentiating higher powers of \( \cot{x} \)?

**Tip**: Always simplify trigonometric identities before differentiating to make the process smoother.

Calculate the derivative of the following functions. (a) f(x) = cot² x

Learn how to calculate the derivative of f(x) = cot²(x) using the chain rule and trigonometric derivatives. This step-by-step solution explains the application of the chain rule and provides insight into differentiating trigonometric functions. Suitable for college-level calculus students.

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