https://file.aiquickdraw.com/mathbot/file/17270119361292tg53oxo.jpg

The image contains three derivative problems related to trigonometric functions, and you're being asked to verify (or "buktikan") the derivatives. Here's a step-by-step breakdown of each problem:

1. **\( y = \cos u \)**
   - Derivative:
     \[
     y' = -\sin u
     \]
   - According to the image, it's stated:
     \[
     y' = \cot u \cdot \cos u
     \]
   We'll need to verify if this is true by finding the derivative of \( \cos u \) and comparing it to the given result.

2. **\( y = \sec u \)**
   - Derivative:
     \[
     y' = \sec u \cdot \tan u
     \]
   This matches the standard derivative rule for \( \sec u \).

3. **\( y = \cot u \)**
   - Derivative:
     \[
     y' = -\csc^2 u
     \]
   This matches the standard derivative rule for \( \cot u \).

### Verifying Each:

- For \( y = \cos u \), the derivative \( y' \) should be \( -\sin u \). The given form \( \cot u \cdot \cos u \) doesn't immediately match. We can further analyze or rewrite to see if there's an identity transformation.
  
Would you like me to further break down the proof steps for each derivative? 

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### Related Questions:
1. How do you compute the derivative of \( \sin u \)?
2. What is the geometric interpretation of trigonometric derivatives?
3. Can the chain rule be applied to any of these derivatives?
4. Why is the derivative of \( \sec u \) written as \( \sec u \cdot \tan u \)?
5. How would the result change if \( u \) were a function of another variable, e.g., \( u(x) \)?

### Tip:
When differentiating trigonometric functions, it's useful to recall key identities and rules like the product rule, chain rule, and quotient rule for more complex forms.

Prove the derivatives for the following trigonometric functions: y = cos(u), y = sec(u), and y = cot(u).

This page provides a step-by-step guide to proving the derivatives of key trigonometric functions: cos(u), sec(u), and cot(u). It includes the standard differentiation rules for trigonometric functions and the verification of each derivative. Suitable for students in Grades 11-12.

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