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Tentukan nilai dy/dt dengan aturan rantai, nyatakan hasilnya dalam t.

The given function is:

\[
y = \frac{x^2}{x^2 + 1}, \quad x = \sqrt{3t + 1}
\]

We are tasked with finding \(\frac{dy}{dt}\) using the chain rule.

### Step-by-step process:

1. **Find \(\frac{dy}{dx}\)**: Differentiate \(y = \frac{x^2}{x^2 + 1}\) with respect to \(x\).
   
   Using the quotient rule:

   \[
   \frac{dy}{dx} = \frac{(x^2 + 1) \cdot 2x - x^2 \cdot 2x}{(x^2 + 1)^2}
   \]
   Simplify the numerator:
   \[
   \frac{dy}{dx} = \frac{2x(x^2 + 1 - x^2)}{(x^2 + 1)^2} = \frac{2x}{(x^2 + 1)^2}
   \]

2. **Find \(\frac{dx}{dt}\)**: Now, differentiate \(x = \sqrt{3t + 1}\) with respect to \(t\).
   
   \[
   \frac{dx}{dt} = \frac{1}{2} \cdot (3t + 1)^{-\frac{1}{2}} \cdot 3 = \frac{3}{2\sqrt{3t + 1}}
   \]

3. **Apply the chain rule**: Now, use the chain rule to find \(\frac{dy}{dt}\):
   
   \[
   \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}
   \]
   Substituting the expressions for \(\frac{dy}{dx}\) and \(\frac{dx}{dt}\):
   
   \[
   \frac{dy}{dt} = \frac{2x}{(x^2 + 1)^2} \cdot \frac{3}{2\sqrt{3t + 1}}
   \]
   Simplify:
   \[
   \frac{dy}{dt} = \frac{3x}{(x^2 + 1)^2 \cdot \sqrt{3t + 1}}
   \]
   
4. **Substitute \(x = \sqrt{3t + 1}\) into the result**:

   Since \(x = \sqrt{3t + 1}\), substitute this into the expression:
   
   \[
   \frac{dy}{dt} = \frac{3\sqrt{3t + 1}}{((\sqrt{3t + 1})^2 + 1)^2 \cdot \sqrt{3t + 1}}
   \]
   Simplify further:
   
   \[
   \frac{dy}{dt} = \frac{3}{(3t + 1 + 1)^2} = \frac{3}{(3t + 2)^2}
   \]

Thus, the value of \(\frac{dy}{dt}\) is:

\[
\frac{dy}{dt} = \frac{3}{(3t + 2)^2}
\]

Let me know if you would like further clarification or a more detailed explanation.

Here are some related questions you might find helpful:
1. How does the quotient rule work in differentiation?
2. What is the chain rule, and when is it used?
3. Can we find higher-order derivatives of the given function \(y\)?
4. How does implicit differentiation differ from the chain rule?
5. Can we apply a different method, like substitution, to solve this?

**Tip:** When differentiating composite functions, always identify the outer and inner functions clearly to apply the chain rule correctly.

Learn how to apply the chain rule to find dy/dt for the function y = (x^2)/(x^2 + 1) and x = √(3t + 1). This step-by-step solution covers key calculus concepts including the quotient and chain rule, suitable for university-level calculus students.

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