Find the derivative of the following functions.
(i) `((x + 1)/(x + 2))`
(ii) `(x^(2) + 1)/("sin" x)`
(iii) `(e^(x))/(1 + cos x)`

Explanation

To find the derivatives of the given functions, we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form of \( \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), then the derivative is given by:

\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2} \]

where \( u' \) is the derivative of \( u \) and \( v' \) is the derivative of \( v \).

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