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To differentiate the expression \((x + 2)(x - 1)\) with respect to \(x\) using the product rule, let's proceed step-by-step.

Given:
\[
f(x) = (x + 2)(x - 1)
\]

The product rule states:
\[
\frac{d}{dx}[u \cdot v] = u'v + uv'
\]
where \(u = x + 2\) and \(v = x - 1\).

### Step 1: Differentiate each part separately.
1. \(u = x + 2 \Rightarrow u' = 1\).
2. \(v = x - 1 \Rightarrow v' = 1\).

### Step 2: Apply the product rule.
Using the product rule:
\[
\frac{d}{dx}[(x + 2)(x - 1)] = (x + 2)(1) + (x - 1)(1)
\]

### Step 3: Simplify the expression.
\[
= (x + 2) + (x - 1) = x + 2 + x - 1 = 2x + 1
\]

### Final Answer:
\[
\frac{d}{dx}[(x + 2)(x - 1)] = 2x + 1
\]

Would you like further details, or do you have any questions?

Here are some related questions to explore:
1. How does the product rule work for functions that are more complex?
2. What is the chain rule, and how does it differ from the product rule?
3. Can we apply the quotient rule similarly for division of functions?
4. How can we verify this derivative using the expanded form of \((x+2)(x-1)\)?
5. What would happen if we had a constant multiplier in front of the expression?

**Tip:** Always simplify expressions as much as possible before differentiating to make calculations easier.

Derive (x + 2)(x - 1) using the product rule.

Learn how to differentiate the expression (x + 2)(x - 1) using the product rule. This step-by-step solution covers differentiation techniques, including finding derivatives of polynomial functions. Suitable for students in Grades 11-12.

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