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The problem asks to find the first and second derivatives of the function:

\[
f(x) = \frac{x^2 - 1}{x^2 + 1}
\]

Let's go through each step.

### Step 1: Find \( f'(x) \)

To differentiate \( f(x) = \frac{x^2 - 1}{x^2 + 1} \), we use the **quotient rule**, which states that if \( f(x) = \frac{g(x)}{h(x)} \), then:
\[
f'(x) = \frac{g'(x) h(x) - g(x) h'(x)}{[h(x)]^2}
\]

In this case:
- \( g(x) = x^2 - 1 \) and \( h(x) = x^2 + 1 \)
- \( g'(x) = 2x \) and \( h'(x) = 2x \)

Applying the quotient rule:
\[
f'(x) = \frac{(2x)(x^2 + 1) - (x^2 - 1)(2x)}{(x^2 + 1)^2}
\]

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

Expanding each term in the numerator:
\[
= \frac{(2x^3 + 2x) - (2x^3 - 2x)}{(x^2 + 1)^2}
\]

Combine like terms:
\[
= \frac{2x^3 + 2x - 2x^3 + 2x}{(x^2 + 1)^2} = \frac{4x}{(x^2 + 1)^2}
\]

So, the first derivative is:
\[
f'(x) = \frac{4x}{(x^2 + 1)^2}
\]

### Step 2: Find \( f''(x) \)

To find the second derivative \( f''(x) \), we need to differentiate \( f'(x) = \frac{4x}{(x^2 + 1)^2} \) again, using the **quotient rule**.

Let:
- \( g(x) = 4x \)
- \( h(x) = (x^2 + 1)^2 \)

Then:
- \( g'(x) = 4 \)
- \( h'(x) = 2(x^2 + 1)(2x) = 4x(x^2 + 1) \)

Applying the quotient rule again:
\[
f''(x) = \frac{g'(x) h(x) - g(x) h'(x)}{[h(x)]^2}
\]
\[
= \frac{4(x^2 + 1)^2 - 4x \cdot 4x(x^2 + 1)}{(x^2 + 1)^4}
\]

Simplify this expression:
\[
= \frac{4(x^2 + 1)^2 - 16x^2(x^2 + 1)}{(x^2 + 1)^4}
\]

Factor out \( 4(x^2 + 1) \) from the numerator:
\[
= \frac{4(x^2 + 1)(x^2 + 1 - 4x^2)}{(x^2 + 1)^4}
\]

Simplify inside the parentheses:
\[
= \frac{4(x^2 + 1)(1 - 3x^2)}{(x^2 + 1)^3}
\]

So, the second derivative is:
\[
f''(x) = \frac{4(1 - 3x^2)}{(x^2 + 1)^3}
\]

### Final Answer:

1. First derivative: \( f'(x) = \frac{4x}{(x^2 + 1)^2} \)
2. Second derivative: \( f''(x) = \frac{4(1 - 3x^2)}{(x^2 + 1)^3} \)

Would you like more details on any step?

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Here are some related questions to explore:

1. How can the quotient rule be used in real-life applications?
2. What is the physical significance of the second derivative in motion?
3. How would you find higher-order derivatives for this function?
4. How does the behavior of the function change at critical points of \( f'(x) \)?
5. Can \( f(x) \) have any points of inflection, and how would you determine them?

**Tip:** Remember, the quotient rule is particularly useful for rational functions but can often be avoided by simplifying expressions first, if possible.

If f(x) = (x^2 - 1)/(x^2 + 1), find f'(x) and f''(x).

This guide demonstrates how to find the first and second derivatives of f(x) = (x^2 - 1)/(x^2 + 1) using the quotient rule. Learn step-by-step differentiation techniques for solving rational functions, suitable for students in Grades 11-12.

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