To find the derivative of \((u^2 - 1)(u^3 + 1)\), we'll apply the **product rule**, which states:

\[
\frac{d}{du} [f(u)g(u)] = f'(u)g(u) + f(u)g'(u)
\]

Let \(f(u) = u^2 - 1\) and \(g(u) = u^3 + 1\).

### Step 1: Find \(f'(u)\)
\[
f'(u) = \frac{d}{du}(u^2 - 1) = 2u
\]

### Step 2: Find \(g'(u)\)
\[
g'(u) = \frac{d}{du}(u^3 + 1) = 3u^2
\]

### Step 3: Apply the product rule
Now we substitute into the product rule formula:

\[
\frac{d}{du}[(u^2 - 1)(u^3 + 1)] = f'(u)g(u) + f(u)g'(u)
\]

\[
= (2u)(u^3 + 1) + (u^2 - 1)(3u^2)
\]

### Step 4: Simplify
Distribute the terms:

\[
= 2u(u^3 + 1) + (u^2 - 1)(3u^2)
\]

\[
= 2u^4 + 2u + 3u^4 - 3u^2
\]

### Step 5: Combine like terms
\[
= (2u^4 + 3u^4) + 2u - 3u^2
\]

\[
= 5u^4 - 3u^2 + 2u
\]

### Final Answer:
\[
\frac{d}{du}[(u^2 - 1)(u^3 + 1)] = 5u^4 - 3u^2 + 2u
\]

Do you have any questions or would you like further clarification?

Here are 5 related questions to practice:
1. What is the derivative of \((u^3 - 2)(u^2 + 3)\)?
2. Find the derivative of \((u^2 + 1)(u^4 - 4)\).
3. How does the product rule work for three functions \((f(u)g(u)h(u))\)?
4. Derive \((x^2 + x)(x^3 + x)\) using the product rule.
5. What is the second derivative of \((u^2 - 1)(u^3 + 1)\)?

**Tip:** When applying the product rule, always differentiate one function at a time, leaving the other function intact.

Learn how to find the derivative of (u^2 - 1)(u^3 + 1) step-by-step using the product rule of differentiation. This calculus problem demonstrates key concepts like the product rule, differentiation of polynomials, and simplification. Suitable for Grades 11-12 or introductory college calculus students.

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