To differentiate $(x^2 - 1)^n$ with respect to $x$, we use the chain rule.

Let: $u = x^2 - 1$ Then, we have $f(x) = u^n$.

The chain rule says: [ \frac{d}{dx} [f(x)] = \frac{d}{du} [u^n] \cdot \frac{du}{dx} ]

Step 1: Differentiate $u^n$ with respect to $u$

$\frac{d}{du} [u^n] = n u^{n-1}$

Step 2: Differentiate $u = x^2 - 1$ with respect to $x$

$\frac{du}{dx} = 2x$

Step 3: Apply the chain rule

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

Final result:

$\frac{d}{dx} \left( (x^2 - 1)^n \right) = 2nx(x^2 - 1)^{n-1}$

Would you like more details or have any questions?

Here are five related questions:

1. How would you differentiate $(x^2 + 1)^n$?
2. What if $n$ were a function of $x$? How would the differentiation change?
3. How do you apply the chain rule when differentiating nested functions?
4. What is the second derivative of $(x^2 - 1)^n$?
5. How do you differentiate $(x^2 - 1)^n$ with respect to $n$?

Tip: When applying the chain rule, always first identify the inner function and the outer function clearly before differentiating.