To determine what \( f''(x) \) (the second derivative) is equal to, we need the specific function \( f(x) \). However, if we continue from your previous question where:

\[
f(x) = \ln\left(x - \frac{1}{x}\right)
\]

we already calculated the first derivative:

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

Now, let’s find \( f''(x) \), the derivative of \( f'(x) \), using the quotient rule. The quotient rule states:

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

where, for \( f'(x) \), we have:
- \( u = x^2 + 1 \) and \( u' = 2x \)
- \( v = x^3 - x \) and \( v' = 3x^2 - 1 \)

### Step-by-Step Calculation of \( f''(x) \)

Applying the quotient rule:

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

Expanding each term:

1. **First part of the numerator**: \( (2x)(x^3 - x) = 2x^4 - 2x^2 \)

2. **Second part of the numerator**: \( (x^2 + 1)(3x^2 - 1) = 3x

Learn how to calculate the second derivative of a function, f''(x), using the quotient rule. This example demonstrates the step-by-step process of differentiating a complex function involving ln(x - 1/x), making it ideal for calculus students in Grades 11-12.

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