(−x^2+cotx+x^2sinx+cotxsinx+xcsc^2x−2xcosx+cosxcsc^2x​)/(x^2+cotx)^2

Given the simplified derivative of the function \( f(x) = \frac{x - \cos x}{x^2 + \cot x} \), let's rewrite it clearly:

\[ f'(x) = \frac{-x^2 + \cot x + x^2 \sin x + \cot x \sin x + x \csc^2 x - 2x \cos x + \cos x \csc^2 x}{(x^2 + \cot x)^2} \]

Let's break down each term in the numerator to verify and ensure it's correctly combined:

1. \( -x^2 \)
2. \( \cot x \)
3. \( x^2 \sin x \)
4. \( \cot x \sin x \)
5. \( x \csc^2 x \)
6. \( -2x \cos x \)
7. \( \cos x \csc^2 x \)

Combining like terms if possible:

\[ = \frac{-x^2 + x^2 \sin x + \cot x + \cot x \sin x + x \csc^2 x - 2x \cos x + \cos x \csc^2 x}{(x^2 + \cot x)^2} \]

This is the final form of the derivative \( f'(x) \).

Would you like any more details or have any specific questions?

Here are 5 numbered questions you might ask:
1. Can you show how to find the derivative using the chain rule?
2. How do you differentiate a product of functions using the product rule?
3. Can you provide an example of differentiating an implicit function?
4. How do you integrate functions involving trigonometric identities?
5. Can you explain how to find the second derivative of a function?

Tip: When combining terms in a derivative, always ensure that each term is correctly simplified and grouped to avoid any errors.

Understand how to simplify and compute the derivative of a complex trigonometric function. This problem involves differentiation, trigonometric identities, and the chain rule. Suitable for undergraduate students studying calculus.

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