1. **State the problem:** Find the derivative of the function $$f(x) = \frac{\cot x}{1 + \csc x}$$. 2. **Recall formulas and rules:** - Derivative of $$\cot x$$ is $$-\csc^2 x$$. - Derivative of $$\csc x$$ is $$-\csc x \cot x$$. - Use the quotient rule: If $$f(x) = \frac{u(x)}{v(x)}$$, then $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$$. 3. **Identify $$u(x)$$ and $$v(x)$$:** - $$u(x) = \cot x$$ - $$v(x) = 1 + \csc x$$ 4. **Compute derivatives:** - $$u'(x) = -\csc^2 x$$ - $$v'(x) = 0 + (-\csc x \cot x) = -\csc x \cot x$$ 5. **Apply quotient rule:** $$ f'(x) = \frac{(-\csc^2 x)(1 + \csc x) - (\cot x)(-\csc x \cot x)}{(1 + \csc x)^2} $$ 6. **Simplify numerator:** $$ = \frac{-\csc^2 x - \csc^3 x + \cot^2 x \csc x}{(1 + \csc x)^2} $$ 7. **Use identity:** $$\cot^2 x = \csc^2 x - 1$$, so $$ \cot^2 x \csc x = (\csc^2 x - 1) \csc x = \csc^3 x - \csc x $$ 8. **Substitute back:** $$ \text{Numerator} = -\csc^2 x - \csc^3 x + \csc^3 x - \csc x = -\csc^2 x - \csc x $$ 9. **Factor numerator:** $$ -\csc x (\csc x + 1) $$ 10. **Final derivative:** $$ f'(x) = \frac{-\csc x (\csc x + 1)}{(1 + \csc x)^2} = \frac{-\csc x (1 + \csc x)}{(1 + \csc x)^2} = \frac{-\csc x}{1 + \csc x} $$ **Answer:** $$f'(x) = \frac{-\csc x}{1 + \csc x}$$