To find the derivative of the function 

\[
y = \frac{x}{x^2 + 4}
\]

we will use the **quotient rule**, which states that if we have a function in the form \( y = \frac{f(x)}{g(x)} \), then the derivative \( y' \) is given by:

\[
y' = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
\]

Here:
- \( f(x) = x \) and \( g(x) = x^2 + 4 \)
- \( f'(x) = 1 \)
- \( g'(x) = 2x \)

Let's apply the quotient rule:

\[
y' = \frac{(1)(x^2 + 4) - (x)(2x)}{(x^2 + 4)^2}
\]

Now, simplify the expression:

1. Expand the terms in the numerator:

   \[
   y' = \frac{x^2 + 4 - 2x^2}{(x^2 + 4)^2}
   \]

2. Combine like terms:

   \[
   y' = \frac{-x^2 + 4}{(x^2 + 4)^2} = \frac{4 - x^2}{(x^2 + 4)^2}
   \]

So, the derivative of \( y = \frac{x}{x^2 + 4} \) is:

\[
y' = \frac{4 - x^2}{(x^2 + 4)^2}
\]

Would you like additional details or have any questions?

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Here are 5 related questions:
1. How can we find the second derivative of \( y = \frac{x}{x^2 + 4} \)?
2. What is the derivative of \( y = \frac{x}{x^2 + c} \) for a general constant \( c \)?
3. What are the critical points of \( y = \frac{x}{x^2 + 4} \)?
4. How does the derivative change if we add a constant to the numerator?
5. Can we use logarithmic differentiation on this function? If so, how?

**Tip:** The quotient rule can be simplified by rewriting the function if possible, but it's usually best for functions with more complex denominators.

Discover how to find the derivative of y = x/(x^2 + 4) using the quotient rule, a fundamental calculus technique. This solution walks through each step, simplifying the result to y' = (4 - x^2) / (x^2 + 4)^2, and includes related questions for deeper understanding. Ideal for students in Grades 11-12.

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