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.