Derivatives Find and simplify the derivative of the following functions.
g(w) = √w+w / √w-w

Verified step by step guidance

1. Step 1: Rewrite the function \( g(w) = \frac{\sqrt{w} + w}{\sqrt{w} - w} \) in a form that is easier to differentiate. Consider using the quotient rule, which states that if you have a function \( \frac{u}{v} \), its derivative is \( \frac{u'v - uv'}{v^2} \).
2. Step 2: Identify \( u = \sqrt{w} + w \) and \( v = \sqrt{w} - w \). Find the derivatives \( u' \) and \( v' \). For \( u = \sqrt{w} + w \), use the derivative rules: \( u' = \frac{1}{2\sqrt{w}} + 1 \). For \( v = \sqrt{w} - w \), use: \( v' = \frac{1}{2\sqrt{w}} - 1 \).
3. Step 3: Apply the quotient rule: \( g'(w) = \frac{(\frac{1}{2\sqrt{w}} + 1)(\sqrt{w} - w) - (\sqrt{w} + w)(\frac{1}{2\sqrt{w}} - 1)}{(\sqrt{w} - w)^2} \).
4. Step 4: Simplify the expression in the numerator by distributing and combining like terms. Carefully expand each term and simplify.
5. Step 5: Simplify the entire expression by combining like terms and reducing the fraction if possible. Ensure that the final expression is in its simplest form.