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, which states that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{u'v - uv'}{v^2}$.Let $u = x$ and $v = 2x+5$.Then, $u' = \frac{d}{dx}(x) = 1$.And $v' = \frac{d}{dx}(2x+5) = 2$.Substitute these into the quotient rule formula:$\frac{dy}{dx} = \frac{(1)(2x+5) - (x)(2)}{(2x+5)^2}$$= \frac{2x+5 - 2x}{(2x+5)^2}$$= \frac{5}{(2x+5)^2}$.Thus, option D is the correct answer.
Hint
Apply the quotient rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}$.