To differentiate the function \( y = \frac{3}{4 + x^2} \), we will use the quotient rule.

The quotient rule states that if \( y = \frac{u(x)}{v(x)} \), then:
\[
y' = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}
\]
In our case, \( u(x) = 3 \) and \( v(x) = 4 + x^2 \).

### Step 1: Differentiate \( u(x) \) and \( v(x) \)
- \( u'(x) = 0 \) (since \( u(x) = 3 \) is a constant)
- \( v'(x) = 2x \) (since the derivative of \( 4 + x^2 \) is \( 2x \))

### Step 2: Apply the quotient rule
\[
y' = \frac{(4 + x^2)(0) - 3(2x)}{(4 + x^2)^2}
\]
\[
y' = \frac{-6x}{(4 + x^2)^2}
\]

Thus, the derivative of \( y = \frac{3}{4 + x^2} \) is:
\[
y' = \frac{-6x}{(4 + x^2)^2}
\]

Would you like further details on this solution, or any additional questions?

Here are five related questions:
1. How would you differentiate \( y = \frac{1}{4+x^2} \)?
2. What is the derivative of \( y = \frac{3}{4 - x^2} \)?
3. Can you apply the chain rule to differentiate \( y = \frac{1}{x^2} \)?
4. How does the quotient rule compare with the product rule?
5. How would you find the second derivative of \( y = \frac{3}{4+x^2} \)?

**Tip**: Always double-check whether to use the quotient or product rule by identifying the structure of the function first.

Learn how to differentiate y = 3 / (4 + x^2) using the quotient rule in calculus. This step-by-step guide covers the application of the quotient rule, differentiation of constants, and key algebraic simplifications. Suitable for students in Grades 11-12.

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