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To solve the problem and find the second derivative \( f''(x) \) given the function \( f(x) = \frac{3x - 7}{x^2 - 6} \), we'll follow these steps:

### Step 1: Find the first derivative \( f'(x) \).
We'll use the quotient rule to differentiate \( f(x) = \frac{u(x)}{v(x)} \), where \( u(x) = 3x - 7 \) and \( v(x) = x^2 - 6 \).

The quotient rule states:

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

First, find \( u'(x) \) and \( v'(x) \):

\[
u'(x) = \frac{d}{dx}(3x - 7) = 3
\]

\[
v'(x) = \frac{d}{dx}(x^2 - 6) = 2x
\]

Now, apply the quotient rule:

\[
f'(x) = \frac{(x^2 - 6) \cdot 3 - (3x - 7) \cdot 2x}{(x^2 - 6)^2}
\]

Simplify the expression:

\[
f'(x) = \frac{3x^2 - 18 - 6x^2 + 14x}{(x^2 - 6)^2} = \frac{-3x^2 + 14x - 18}{(x^2 - 6)^2}
\]

### Step 2: Find the second derivative \( f''(x) \).
We need to differentiate \( f'(x) \) again. This is also a quotient, so we'll apply the quotient rule again.

Let \( u(x) = -3x^2 + 14x - 18 \) and \( v(x) = (x^2 - 6)^2 \).

First, find \( u'(x) \) and \( v'(x) \):

\[
u'(x) = \frac{d}{dx}(-3x^2 + 14x - 18) = -6x + 14
\]

For \( v(x) = (x^2 - 6)^2 \), apply the chain rule:

\[
v'(x) = 2(x^2 - 6) \cdot 2x = 4x(x^2 - 6)
\]

Now, apply the quotient rule to find \( f''(x) \):

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

Substitute the values:

\[
f''(x) = \frac{(x^2 - 6)^2(-6x + 14) - (-3x^2 + 14x - 18) \cdot 4x(x^2 - 6)}{(x^2 - 6)^4}
\]

This expression simplifies further:

\[
f''(x) = \frac{(x^2 - 6)[(-6x + 14)(x^2 - 6) - 4x(-3x^2 + 14x - 18)]}{(x^2 - 6)^3}
\]

Finally, you can expand and simplify this expression to get the exact form of \( f''(x) \). However, this will be quite complex and likely require careful algebraic manipulation.

Would you like the expanded and simplified form, or do you have any questions?

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Here are some related questions to consider:

1. How do you simplify complex rational expressions in derivatives?
2. What is the physical interpretation of the second derivative?
3. How does the quotient rule differ from the product rule in differentiation?
4. Can you apply the quotient rule directly to trigonometric functions?
5. How does the complexity of finding the second derivative change with different types of functions?

**Tip:** When dealing with second derivatives involving the quotient rule, consider simplifying the first derivative as much as possible before proceeding.

Learn how to find the second derivative f''(x) of the function f(x) = (3x - 7) / (x^2 - 6) using differentiation rules and algebraic manipulation. This problem covers differentiation techniques including the quotient rule and chain rule. Suitable for college-level students studying calculus and mathematical analysis.

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