To find the derivative \( f'(x) \) of the function \( f(x) = \frac{x^2 + 1}{x^2 - 9} \), we can use the quotient rule. For a function in the form \( \frac{g(x)}{h(x)} \), the derivative is:

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

In this case:
- \( g(x) = x^2 + 1 \)
- \( h(x) = x^2 - 9 \)

### Step 1: Find \( g'(x) \) and \( h'(x) \)
1. **Differentiate \( g(x) = x^2 + 1 \):**
   \[
   g'(x) = 2x
   \]

2. **Differentiate \( h(x) = x^2 - 9 \):**
   \[
   h'(x) = 2x
   \]

### Step 2: Substitute into the Quotient Rule
Now, applying the quotient rule:

\[
f'(x) = \frac{(2x)(x^2 - 9) - (x^2 + 1)(2x)}{(x^2 - 9)^2}
\]

### Step 3: Simplify the Expression
Expanding the terms in the numerator:

1. **Expand \( (2x)(x^2 - 9) \):**
   \[
   2x \cdot x^2 - 2x \cdot 9 = 2x^3 - 18x
   \]

2. **Expand \( (x^2 + 1)(2x) \):**
   \[
   2x \cdot x^2 + 2x \cdot 1 = 2x^3 + 2x
   \]

Now, substituting these into the numerator:

\[
f'(x) = \frac{(2x^3 - 18x) - (2x^3 + 2x)}{(x^2 - 9)^2}
\]

Simplify by combining like terms:

\[
f'(x) = \frac{2x^3 - 18x - 2x^3 - 2x}{(x^2 - 9)^2} = \frac{-20x}{(x^2 - 9)^2}
\]

### Final Answer
\[
f'(x) = \frac{-20x}{(x^2 - 9)^2}
\]

Would you like further details or have any questions?

Here are some related questions to consider:
1. How does the quotient rule apply when simplifying fractions?
2. What if we need the second derivative \( f''(x) \)?
3. How can we find the critical points of \( f(x) \)?
4. What happens to \( f'(x) \) when \( x = \pm 3 \)?
5. Can we use limits to confirm the behavior of \( f(x) \) at \( x = \pm 3 \)?

**Tip:** Always simplify carefully after applying the quotient rule to avoid mistakes.

Learn how to find the derivative of f(x) = (x^2 + 1) / (x^2 - 9) by applying the quotient rule. This solution covers key calculus concepts, including derivatives, simplification, and the application of the quotient rule. Suitable for Grade 12 or College Calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation