https://file.aiquickdraw.com/mathbot/file/1729903797192kpmwwfcv.jpg

Let's solve the equation step-by-step.

Given:
\[
y \left( x^2 + 1 \right) = 2
\]

1. **Solve for \( y \):**
   To isolate \( y \), divide both sides by \( x^2 + 1 \):
   \[
   y = \frac{2}{x^2 + 1}
   \]

2. **Calculate \(\frac{dy}{dx}\):**
   Now that we have \( y = \frac{2}{x^2 + 1} \), let's find \(\frac{dy}{dx}\) using the derivative.

   The derivative of \( y = \frac{2}{x^2 + 1} \) can be found using the quotient rule, which states:
   \[
   \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}
   \]
   Here, \( u = 2 \) (a constant) and \( v = x^2 + 1 \).

   - The derivative of \( u \) with respect to \( x \), \( \frac{du}{dx} = 0 \).
   - The derivative of \( v \) with respect to \( x \), \( \frac{dv}{dx} = 2x \).

   Applying the quotient rule:
   \[
   \frac{dy}{dx} = \frac{(x^2 + 1) \cdot 0 - 2 \cdot 2x}{(x^2 + 1)^2} = \frac{-4x}{(x^2 + 1)^2}
   \]

So, the result is:
\[
\frac{dy}{dx} = \frac{-4x}{(x^2 + 1)^2}
\]

Would you like a more detailed explanation, or do you have any questions?

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### Related Questions:
1. How would \(\frac{dy}{dx}\) change if the constant 2 were replaced with another constant \(k\)?
2. Can you solve a similar equation where \(y(x^2 + 1) = x\) instead?
3. How does the quotient rule apply in more complex functions?
4. What happens to \(\frac{dy}{dx}\) as \( x \) approaches infinity?
5. How would you solve this if \(y(x^2 + 1) = 2x\) instead?

### Tip:
Always simplify expressions before differentiating to avoid unnecessary complexity.

Solve the equation y(x^2 + 1) = 2 for y to find an explicit expression for y and then calculate dy/dx.

Find an explicit expression for y from the equation y(x^2 + 1) = 2, and calculate dy/dx using differentiation. This solution covers key calculus concepts like the quotient rule, suitable for advanced high school students in Grades 11-12.

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