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.

Learn how to differentiate (x^2 - 1)^n using the chain rule and power rule. This step-by-step guide covers the core concepts of differentiation, exponentiation, and applying the chain rule, suitable for students in advanced high school calculus or introductory calculus courses.

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