The expression provided for $y$ is ambiguous because of how the division is written. It is unclear whether the entire expression is meant to be divided by $x^2 + 2$, or just the first term.

Option 1: If the expression is $y = \frac{2x}{x^2 + 2}$

In this case, we will apply the quotient rule to find the derivative $\frac{dy}{dx}$.

The quotient rule states that if $y = \frac{f(x)}{g(x)}$, then: $\frac{dy}{dx} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ Here, $f(x) = 2x$ and $g(x) = x^2 + 2$.

1. $f'(x) = 2$
2. $g'(x) = 2x$

Now, applying the quotient rule: $\frac{dy}{dx} = \frac{(2)(x^2 + 2) - (2x)(2x)}{(x^2 + 2)^2}$ Simplifying the numerator:

= \frac{2x^2 + 4 - 4x^2}{(x^2 + 2)^2} = \frac{-2x^2 + 4}{(x^2 + 2)^2}$$ So, the derivative is: $$\frac{dy}{dx} = \frac{-2x^2 + 4}{(x^2 + 2)^2}$$ ### Option 2: If the expression is $$y = \frac{2x}{x^2} + 2$$ This simplifies to: $$y = \frac{2}{x} + 2$$ Now, differentiate term by term: 1. The derivative of $$\frac{2}{x}$$ is $$-\frac{2}{x^2}$$. 2. The derivative of 2 is 0. Thus, the derivative is: $$\frac{dy}{dx} = -\frac{2}{x^2}$$ ### Clarification Needed Which interpretation is correct, Option 1 or Option 2? Let me know so I can confirm the solution! --- Would you like more details on any part of the solution? Here are 5 related questions to consider: 1. How is the quotient rule applied in calculus? 2. What is the difference between implicit and explicit differentiation? 3. How do you handle products and quotients in differentiation? 4. Can the chain rule be used here instead of the quotient rule? 5. What other methods could simplify rational functions before differentiation? **Tip:** Always clarify ambiguous expressions using parentheses, such as $$\frac{2x}{x^2 + 2}$$, to avoid confusion when differentiating.