Question Detail
Question
Given that \(y = \frac{x}{2x+5}\), find \(\frac{dy}{dx}\).
Correct Answer
Option D is the correct answer.
Detailed Explanation
We use the quotient rule for differentiation: (\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}).Let (u = x) and (v = 2x+5).Then (u' = \frac{d}{dx}(x) = 1).And (v' = \frac{d}{dx}(2x+5) = 2).Apply the quotient rule:(\frac{dy}{dx} = \frac{(1)(2x+5) - (x)(2)}{(2x+5)^2})(= \frac{2x+5 - 2x}{(2x+5)^2})(= \frac{5}{(2x+5)^2}).
Hint
Apply the quotient rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}\).