find the y' given the function y= x cos(x) + csc(x)

To find \( y' \) given the function \( y = x \cos(x) + \csc(x) \), we'll differentiate each term separately.

1. **Differentiate \( x \cos(x) \)** using the product rule:
   \[
   \frac{d}{dx} (x \cos(x)) = \frac{d}{dx}(x) \cdot \cos(x) + x \cdot \frac{d}{dx}(\cos(x))
   = \cos(x) - x \sin(x)
   \]

2. **Differentiate \( \csc(x) \)**:
   The derivative of \( \csc(x) \) is \( -\csc(x) \cot(x) \).

Now, putting it all together:
\[
y' = \cos(x) - x \sin(x) - \csc(x) \cot(x)
\]

Thus, the derivative of \( y \) is:
\[
y' = \cos(x) - x \sin(x) - \csc(x) \cot(x)
\]

Would you like more details or have any questions?

Here are five related questions:
1. What is the product rule and how does it apply in differentiation?
2. How do you find the derivative of \( \cos(x) \) and \( \sin(x) \)?
3. What is the derivative of \( \csc(x) \) and how is it derived?
4. How can you interpret the derivative \( y' \) in terms of the slope of the function?
5. How do trigonometric derivatives impact the behavior of functions?

**Tip**: When differentiating products of functions, always apply the product rule, ensuring both terms are properly differentiated.

find the y' given the function y = x cos(x) + csc(x)

Learn how to differentiate the function y = x cos(x) + csc(x) step by step using the product rule and trigonometric derivative formulas. This detailed solution covers key calculus concepts like the product rule and trigonometric derivatives, ideal for students in advanced high school or college calculus.

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