Find the derivative of the trigonometric function y=-csc(x)-cos(x)

We are tasked with finding the derivative of the function:

\[
y = -\csc(x) - \cos(x)
\]

### Step-by-Step Derivation:

1. **Derivative of \( -\csc(x) \):**
   The derivative of \( \csc(x) \) is \( -\csc(x) \cot(x) \). So, the derivative of \( -\csc(x) \) is:
   \[
   \frac{d}{dx}[-\csc(x)] = \csc(x) \cot(x)
   \]

2. **Derivative of \( -\cos(x) \):**
   The derivative of \( \cos(x) \) is \( -\sin(x) \). So, the derivative of \( -\cos(x) \) is:
   \[
   \frac{d}{dx}[-\cos(x)] = \sin(x)
   \]

### Final Result:

The derivative of the function is:

\[
\frac{dy}{dx} = \csc(x) \cot(x) + \sin(x)
\]

Would you like more details on any part of this process?

Here are 5 related questions to deepen your understanding:
1. What is the derivative of \( \sec(x) \)?
2. How is the derivative of \( \tan(x) \) derived?
3. Can you explain how to differentiate products of trigonometric functions?
4. What are the applications of differentiating trigonometric functions?
5. How can you integrate \( \csc(x) \) and \( \cot(x) \)?

**Tip:** Always remember that derivatives of trigonometric functions often involve combinations of other trigonometric functions like \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), etc., so it's essential to memorize these fundamental rules.

Explore the step-by-step process for finding the derivative of the function y = -csc(x) - cos(x). This problem covers key calculus concepts, including the derivative rules for trigonometric functions such as csc(x) and cos(x). Suitable for Grades 11-12 or early college students.

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