Evaluate the derivatives of the following function by the product rule (x^2-4)(x-2)

To find the derivative of the function \((x^2-4)(x-2)\) using the product rule, we'll follow these steps:

### 1. Product Rule
The product rule states that if we have a function \( f(x) = g(x) \cdot h(x) \), then its derivative is given by:
\[
f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)
\]

### 2. Define \( g(x) \) and \( h(x) \)
Let:
\[
g(x) = x^2 - 4
\]
\[
h(x) = x - 2
\]

### 3. Find \( g'(x) \) and \( h'(x) \)
Compute the derivatives of \( g(x) \) and \( h(x) \).

- For \( g(x) = x^2 - 4 \):
\[
g'(x) = 2x
\]

- For \( h(x) = x - 2 \):
\[
h'(x) = 1
\]

### 4. Apply the Product Rule
Using the product rule, we have:
\[
f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)
\]
Substitute \( g'(x) \), \( g(x) \), \( h'(x) \), and \( h(x) \) into the formula:

\[
f'(x) = (2x) \cdot (x - 2) + (x^2 - 4) \cdot (1)
\]

### 5. Simplify the Expression
Now, simplify each term:

- \( (2x) \cdot (x - 2) = 2x^2 - 4x \)
- \( (x^2 - 4) \cdot (1) = x^2 - 4 \)

So, the derivative is:

\[
f'(x) = 2x^2 - 4x + x^2 - 4
\]

Combine like terms:

\[
f'(x) = 3x^2 - 4x - 4
\]

### Final Answer
The derivative of the function \((x^2-4)(x-2)\) is:

\[
f'(x) = 3x^2 - 4x - 4
\]

Would you like more details or have any other questions?

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**Related Questions:**

1. How would you find the derivative of \( (x^2 - 4)(x + 3) \) using the product rule?
2. What happens if you apply the quotient rule to \( \frac{x^2 - 4}{x - 2} \)?
3. Can you find the second derivative of \( (x^2-4)(x-2) \)?
4. How would you find the critical points of the function \( f(x) = (x^2-4)(x-2) \)?
5. What is the derivative of \( (x^2-4)^2 \) using the chain rule?

**Tip:** When using the product rule, always carefully identify and differentiate both functions involved before substituting into the rule.

Learn how to find the derivative of the function (x^2-4)(x-2) using the product rule. This detailed solution walks you through each step of applying the product rule to compute derivatives, simplifying the final expression. Suitable for high school calculus students.

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