To find the derivative of the function \( f(x) = \frac{x - \cos x}{x^2 + \cot x} \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function \( f(x) = \frac{g(x)}{h(x)} \), then its derivative is given by:

\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \]

In our case, \( g(x) = x - \cos x \) and \( h(x) = x^2 + \cot x \).

First, let's find the derivatives of \( g(x) \) and \( h(x) \).

1. \( g(x) = x - \cos x \)

\[ g'(x) = \frac{d}{dx}(x) - \frac{d}{dx}(\cos x) \]
\[ g'(x) = 1 + \sin x \]

2. \( h(x) = x^2 + \cot x \)

\[ h'(x) = \frac{d}{dx}(x^2) + \frac{d}{dx}(\cot x) \]
\[ h'(x) = 2x - \csc^2 x \]

Now, applying the quotient rule:

\[ f'(x) = \frac{(1 + \sin x)(x^2 + \cot x) - (x - \cos x)(2x - \csc^2 x)}{(x^2 + \cot x)^2} \]

Simplifying the numerator:

\[ f'(x) = \frac{(1 + \sin x)(x^2 + \cot x) - (x - \cos x)(2x - \csc^2 x)}{(x^2 + \cot x)^2} \]
\[ = \frac{(x^2 + \cot x + x^2 \sin x + \cot x \sin x) - (2x^2 - x \csc^2 x - 2x \cos x + \cos x \csc^2 x)}{(x^2 + \cot x)^2} \]
\[ = \frac{x^2 + \cot x + x^2 \sin x + \cot x \sin x - 2x^2 + x \csc^2 x - 2x \cos x + \cos x \csc^2 x}{(x^2 + \cot x)^2} \]

Combining like terms:

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

This is the simplified form of the derivative. 

Would you like further details or have any questions?

Here are 5 numbered questions you might ask:
1. Can you show a step-by-step solution to a different derivative problem?
2. How do you apply the product rule for differentiation?
3. Can you explain how to find the derivative of a trigonometric function?
4. What is the chain rule and how is it used in differentiation?
5. How do you solve integrals involving trigonometric functions?

Tip: When finding the derivative of a quotient, always identify the functions \( g(x) \) and \( h(x) \) first and then apply the quotient rule carefully.

Learn how to find the derivative of the function f(x) = (x - cos(x)) / (x^2 + cot(x)) using the quotient rule for differentiation. This problem involves differentiation techniques, including the application of the quotient rule and handling trigonometric functions. Suitable for advanced high school students and beyond.

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