To find the second derivative of the function $f(x) = \left( x^2 - 1 \right)^{\frac{2}{3}}$, we proceed step-by-step:

Step 1: First derivative

Let $f(x) = \left( x^2 - 1 \right)^{\frac{2}{3}}$. Using the chain rule for derivatives, we differentiate this expression.

$f'(x) = \frac{2}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} \cdot \frac{d}{dx}(x^2 - 1)$

The derivative of $x^2 - 1$ is $2x$, so:

$f'(x) = \frac{2}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} \cdot 2x$

Simplifying:

$f'(x) = \frac{4x}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}}$

Step 2: Second derivative

Now, we differentiate $f'(x)$ again using the product rule:

$f''(x) = \frac{d}{dx} \left( \frac{4x}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} \right)$

Using the product rule, we have:

$f''(x) = \frac{4}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} + \frac{4x}{3} \cdot \frac{d}{dx} \left( \left( x^2 - 1 \right)^{-\frac{1}{3}} \right)$

We already know the derivative of $\left( x^2 - 1 \right)^{-\frac{1}{3}}$, using the chain rule:

$\frac{d}{dx} \left( \left( x^2 - 1 \right)^{-\frac{1}{3}} \right) = -\frac{1}{3} \left( x^2 - 1 \right)^{-\frac{4}{3}} \cdot 2x$

Substituting this into the expression for $f''(x)$:

$f''(x) = \frac{4}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} - \frac{4x^2}{3} \cdot \frac{2}{3} \left( x^2 - 1 \right)^{-\frac{4}{3}}$

Simplifying:

$f''(x) = \frac{4}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} - \frac{8x^2}{9} \left( x^2 - 1 \right)^{-\frac{4}{3}}$

Thus, the second derivative is:

$f''(x) = \frac{4}{3} \left( x^2 - 1 \right)^{-\frac{1}{3}} - \frac{8x^2}{9} \left( x^2 - 1 \right)^{-\frac{4}{3}}$

Would you like more details or clarifications?

Here are 5 related questions:

1. How do you apply the product rule in differentiation?
2. What is the chain rule and how is it used in this context?
3. How can we interpret the second derivative graphically?
4. What happens to the derivative at $x = \pm 1$?
5. How can we simplify the second derivative further?

Tip: Always check for common factors in derivatives to simplify expressions before proceeding to the next differentiation step.