https://file.aiquickdraw.com/mathbot/file/1729475164575ooev9xbv.jpg

The function in the image is:

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

You are tasked with finding the first and second derivatives, \( f'(x) \) and \( f''(x) \), and then evaluating them at \( x = 0 \) and \( x = 9 \).

### Step 1: First Derivative \( f'(x) \)

The function \( f(x) = \frac{x^2}{2 + x^2} \) is a quotient, so we will use the **quotient rule**:

\[
\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{u'(x) v(x) - u(x) v'(x)}{v(x)^2}
\]

Where \( u(x) = x^2 \) and \( v(x) = 2 + x^2 \).

- \( u'(x) = 2x \)
- \( v'(x) = 2x \)

Now applying the quotient rule:

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

Simplifying the numerator:

\[
f'(x) = \frac{2x(2 + x^2) - 2x(x^2)}{(2 + x^2)^2} = \frac{2x(2 + x^2 - x^2)}{(2 + x^2)^2} = \frac{4x}{(2 + x^2)^2}
\]

### Step 2: Second Derivative \( f''(x) \)

Now we need to find the second derivative of \( f'(x) = \frac{4x}{(2 + x^2)^2} \). We'll apply the **quotient rule** again, where:

- \( u(x) = 4x \)
- \( v(x) = (2 + x^2)^2 \)

- \( u'(x) = 4 \)
- \( v'(x) = 2(2 + x^2)(2x) = 4x(2 + x^2) \)

Now, applying the quotient rule again:

\[
f''(x) = \frac{4(2 + x^2)^2 - 4x \cdot 4x(2 + x^2)}{(2 + x^2)^4}
\]

Simplifying the numerator:

\[
f''(x) = \frac{4(2 + x^2)^2 - 16x^2(2 + x^2)}{(2 + x^2)^4}
\]

We can factor out \( 4(2 + x^2) \) from the numerator:

\[
f''(x) = \frac{4(2 + x^2) \left( (2 + x^2) - 4x^2 \right)}{(2 + x^2)^4}
\]

Simplify the term in parentheses:

\[
f''(x) = \frac{4(2 + x^2)(2 - 3x^2)}{(2 + x^2)^4}
\]

Thus:

\[
f''(x) = \frac{4(2 - 3x^2)}{(2 + x^2)^3}
\]

### Step 3: Evaluate \( f''(x) \) at \( x = 0 \) and \( x = 9 \)

#### \( f''(0) \):

Substitute \( x = 0 \) into \( f''(x) \):

\[
f''(0) = \frac{4(2 - 3(0)^2)}{(2 + 0^2)^3} = \frac{4(2)}{2^3} = \frac{8}{8} = 1
\]

#### \( f''(9) \):

Substitute \( x = 9 \) into \( f''(x) \):

\[
f''(9) = \frac{4(2 - 3(9)^2)}{(2 + 9^2)^3} = \frac{4(2 - 243)}{(2 + 81)^3} = \frac{4(-241)}{83^3}
\]

Now, calculate \( 83^3 = 571787 \):

\[
f''(9) = \frac{-964}{571787}
\]

Thus, \( f''(9) \approx -0.00169 \).

### Final Results:
- \( f''(0) = 1 \)
- \( f''(9) \approx -0.00169 \)

Would you like more details on any step, or have any questions? Here are some follow-up questions to expand on this:

1. How is the quotient rule applied to functions like this?
2. What happens to the higher-order derivatives if the function changes slightly?
3. Can we estimate \( f''(x) \) at values near 9 without using exact calculations?
4. What are other methods for checking the concavity of \( f(x) \)?
5. How does the second derivative relate to the graph's curvature?

**Tip**: Always simplify expressions before differentiating multiple times to avoid excessive complexity.

For the function f(x) = x^2 / (2 + x^2), find f'(x). Then find f''(0) and f''(9).

This problem covers the application of the quotient rule to find the first and second derivatives of the function f(x) = x^2 / (2 + x^2). It also includes the evaluation of f''(x) at x = 0 and x = 9. Suitable for calculus students in Grades 11-12.

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